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IN WHICH IS EXPLAINED AND APPLIED TO PRACTICAL PURPOSES, IN ADDITION
TO THE ORDINARY RULES OF OPERATION,
THE PRINCIPLE OF CANCELING;
DESIGNED FOR SCHOOLS AND ACADEMIES.
BY C. TRACY, A. M.
PRINCIPAL OF NORWICH ACADEMY.
PUBLISHED BY DURRIE & PE C K.
NEW YORK: COLLINS, KEESE, & Co.
BOSTON: CROCKER & BREWSTER.
DURRIE & PECK,
HITCHCOCK AND STAFFORD, PRINTERS,
It will readily be conceded, that all efforts in behalf of the general diffusion of useful knowledge, are in themselves commendable. There is, however, and probably ever will be, a difference of opinion relative to the extent to which books of any particular description, and treating upon the same general topic, may be multiplied, and the interests of education uniformly advanced thereby.
This difference of opinion exists especially in relation to books designed for the use of common schools and academies, and which treat upon the more common subjects of study. The multiplication of books of this description, to the extent realized at the present day, is regarded by many as injurious to the general good. That its tendency is to increase, in some small degree, the expense of education, at least in some parts of the country, will not be denied. But before sentence of final condemnation is pronounced, it always becomes those, who sit as umpires, to take as extended views of the subject before them, as the nature of the case will admit,
The question now presented is, how far the general good is advanced by the multiplication of school books.
To answer this, let it be supposed that only a single work in each department of science studied in common schools, had ever been presented to the public, and that each work were such as it should be. Books of this description would obviously find ample circulation, sufficient, perhaps, to satisfy both authors and publishers, without embracing one half of the ground to be occupied. The consequence would be, that the more recently settled parts of our country would be but poorly supplied with the means of education, for at least some considerable period of time. But as it now is, with such a multiplicity of school books constantly emanating from the press, a spirit of rivalry is created, a desire is excited on the part of both authors and publishers to give to their several works a more extended circulation than can be obtained without exploring the whole ground. As a natural consequence, the inhabitant of the less favored portion of our land, is scarcely settled in his log cabin, before books of every description necessary for the education of his sons and daughters, are presented him, as it were, at his own door. His attention is thus directed to a subject second in importance to none of a temporal nature; and one which, when duly presented, will be likely to be regarded, and to receive a consideration, which otherwise might be long neglected.
The truth of our supposition, that any one set of school books is such in all respects as is required, may, however, very reasonably be doubted. Many of them are unquestionably of a very high order, and very probably owe some degree of their merit to the fact that other minds have been, are, and yet will be, traversing the same ground which their authors trod, and are preparing other works, to supersede them, if possible, in the estimation of the public.
The effect of the multiplication of school books is, therefore, to render the means of education as perfect as the nature of their subjects will allow, and to convey these means, thus perfected, to every part of our entire country.
From the preceding considerations, the author is inclined to regard the multiplication of school books as favorable to the cause of general education. It therefore only remains to point out some of the more important features of the following work, before introducing it to the ordeal of public opinion. That it is worthy of public attention and patronage, belongs not to him to decide. It certainly will be found to possess some peculiarities, which are of course regarded by him as improvements. Whether they are indeed such, remains for others to determine.
A peculiar feature of the following pages, and one which distinguishes this work from every other on the same subject, is the "System of Canceling," which, in connection with the ordinary mode of solution, is introduced throughout, and applied to such arithmetical problems as embrace in their operation both multiplication and divis. ion. This is regarded by the author, and by many others acquainted with his system,
as a decided improvement upon all Arithmetics heretofore presented to the public. The following are some of the advantages of the new system :
1st. The statement required, or, rather, recommended for canceling, is itself a complete analysis of the sum proposed. Suppose, for illustration, that 12 yards of cloth cost $48, and that it is required to find the value of 15 yards of the same.
.We analyze the preceding sum, either by first finding the value of one yard of the cloth, viz. $48• 12 yd. = $4, and then multiplying that price by the number of yards, as, $4X15 yd. $60, Ans.; or by finding the ratio of the number of yards of which the price is given, and of those of which the price is required, and then multiplying that ratio by the given cost. This ratio is l=; and 4x48=60, the number of dollars required, and the same as above. The same stated for canceling:
, and 15X4=60, the dollars required.
The application of the canceling principle is, however, more completely illustrated by the solution of sums in which the ratio of one of The given quantities to a required quantity of the same kind, is traced through several simple ratios. The following sum may serve as an illustration :
If 3 men, in 16 days, of 9 hours each, build a wall 20 feet long, 6 feet high, and 4 feet thick, in how many days, of 8 hours each, will 12 men build a wall 200 feet long, 8 feet high, and 6 feet thick ?
It is obvious the ratio of the given days to the required number of days, is compounded of the ratios, 8 hours : 9 hours; 12 men : 3 men; 20 feet in length: 200 feet in length ; 6 feet in height: 8 feet in height; and 4 feet in thickness: 6 feet in thickness. Or, these ratios may be fractionally expressed, thus : g, ia, 2000, X, and
Now the given days of 9 hours each are changed to days of 8 hours each, by the following statement:
8 = 18 days.
The time, in days of 8 hours each, required for the 12 men to complete the work of 3 men, is obtained by uniting the second of the preceding ratios to the above statement; thus :
16. 9. 3
8. 12 By introducing into the same statement, the third of the preceding ratios, we obtain the requisite time for completing the 200 feet of wall, allowing the height and thickness of each wall to be the same; thus:
16. 9. 3. 200
= 45 days.
8. 12. 20 If the fourth ratio be united, we obtain the time required, allowing each wall to be of the same thickness ; thus: 16. 9. 3. 200. 8
8. 12. 20.6 Lastly, if the fifth and last ratio be introduced, the number of days required by all the conditions of the question, is obtained ; viz. 16. 9. 3. 200. 8. 6
=the required days.
8. 12. 20. 6. 4 The last statement canceled :
10 % 18. 9. 3. 200. 8. &
and 10X9=90 days, answer. 8. 12. 20. 6. 4'
% 2dly. A second advantage to be derived from the canceling system, is the facility afforded by it for reducing several operations to a single statement. The following example will afford an illustration :
Bought 742 lb. of wool, a deduction of 5 per cent from the gross weight being made, for dust, &c. For the net weight, I paid 9 s. New York currency, per lb., and for ready money, was allowed a deduction