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system gave new interest to the scholar; that it afforded an almost immediate solution to questions of the most difficult character; that it greatly abridged the labor of operation, in all cases where it could be introduced, and at all times carried satisfactory evidence of its perfect accuracy in principle, the author of this Arithmetic has expended much time and labor in extending this mode of operation to all arithmetical rules involving proportion, and in giving full explanations and demonstrations of its accuracy of application.
A number of teachers are now employed in teaching upon this principle, whom the author instructed to qualify them for this purpose ; and they all concur in the advantages of the system.
The principle upon which cancelling proceeds is plain. It is merely taking from equal balanced scales equal weights, leaving the balance undisturbed. As applied in arithmetic, each process greatly simplifies the expression of the proportion.
In Fractions, le is $, or , or 1. Here each term of the fraction is equally diminished, and the fraction remains the same in value, but the expression is greatly simplified. By stating a proportional question, and cancelling from it all redundancies, the answer is alone left, or so few portions of the statement remain, that, by a perfectly easy combination, in the manner presented by the rule, the answer is at once obtained.
There is an interest attending the statement of a question upon this system which does not occur in other modes. All fractions in the processes of Multiplication aud Division are treated as whole numbers; the numerator being the same as the dividend in Division, and the denominator the same as the divisor. The facility of passing from one denomination to another, and of performing the reduction, when the terms are of different denominations, is of great advantage. The cancelling is an amusement, and the necessary combination of numbers, to close the question is ordinarily one of the simplest arithmetical operations. The scholar finds that the most difficult questions yield readily to his comprehension of them, and to this mode of solution, and he has the satisfaction of proceeding upon principles which he knows to be unerring.
The system of cancelling is applicable, as stated in the
title page of this work, to all Proportional Questions, embracing the Rules of Three, Single and Double, Direct and Inverse ; Interest ; Discount ; Barter ; Loss and Gain ; Exchange; Reduction, Multiplication and Division of Fractions, &c.
On the 129–30–31–35–36 and 137th pages, and in many other parts of the work, are questions shewing the mode of operation, as compared with the rules in ordinary use ; some of these are selected from other works, for the purpose of illustration. On the 121–2–3–4 and 125th pages, are questions worked out, and the principle explained and illustrated.
Questions of the following kind are found in most Arithmetics, viz.
1. If 3 men can build 360 rods of wall in 24 days, how many rods can 8 men build in 27 days?
2. If a man can travel 240 miles in 12 days, when the days are 12 hours lorg, how far can he travel in 27 days, when the days are 16 hours long ?
3. A merchant bartered 5% cwt. of sugar, at 6 d. per pound, for tea at 8 gs. per pound. How much tea did he receive ? 4. If 500 men
consume 10254 barrels of four in 9 months, how many barrels will 365 men consume in the same time?
These questions are wrought out on the pages first referred to; and all similar questions can be stated and solved by this system, with a facility vastly beyond that of proceeding by the ordinary rules. It is of peculiar importance in solving questions which arise in the daily transactions of business men ; and the rules of the system are so simple, that when they have been once practiced, they can never be supplanted from the memory.
It would be useless to make these remarks, unless fully convinced of their accuracy,
We are aware that in submitting this work to the public, so far as it is introduced, it would be a serious detriment to it, to claim for it any thing more than would be readily conceded by instructors and pupils.
It is submitted to them, with an earnest desire that it may meet their approbation, and with no other object than to aid in promoting this important branch of science.
A mere casual inspection of an arithmetic will of course
furnish no evidence of its utility. It should be taken up systematically, and its mode of operation tested. We commend it to those who will bestow this labor upon it, and should regret to learn that they have reason to dissent from us.
The ordinary rules are inserted throughout, side by side, with the system of cancelling in those rules where that mode of operation is adopted ; and it is believed that this portion of the work will compare favorably with other arithmetics. Many of the rules must be uniform in all systems, and no system of arithmetic can dispense with mechanical labor. All the primary rules preparatory to those involving proportions are in all cases essential, and the science of numbers in its best estate will require skill and labor.
Arithmetical computation and combination manifest a high advance in civilization. Barbarous nations have no idea of numbers beyond a very limited extent, and all results in science, obtained by calculation, are wrought out on the simple principles of arithmetic. Without her aid, Astronomy, Geography and Navigation would be a mere isolated detail of a few barren facts, and Commerce would be utterly at an end.
All conclusions, based upon relation of space and distance, till discoveries have been extended to the utmost verge of creation, have been achieved by arithmetic, attending as the handmaid of science.
Its power, as a means of high attainment and mental discipline, and its practical utility, should command attention to its study and interest in its instruction. It should be second to no other branch in either department of male or female education.
CHARLES G. BURNHAM. PEMBROKE, N. H., OCTOBER 23, 1837.
The author of this work acknowledges himself indebted to Mr. HENRY L. BURNHAM, for many original arithmetical questions, and also for important aid in facilitating the preparation of the work for
Commission, Brokerage and Insurance,
164 Compound Interest,
168 Bank Discount,
169 Loss and Gain,
175 Supplement to Interest, Discount, &c.,
178 Equation of Payments, Fellowship,
181 Assessment of Taxes,
183 Double Fellowship,
188 Extraction of the Square Root, 189 Extraction of the Cube Root,
195 Extraction of Roots in general, 200 Arithmetical Progression,
201 Geometrical Progression,
Compound Interest by Progression, 208
214 Single Position,
- 215 Double Position,
216 Alligation Medial,
219 Alligation Alternate,
224 Miscellaneous Rules,
227 Mensuration of Surfaces,
227 Mensuration of Solids,
232 Measuring Grain,
233 Of the Lever,
234 Of the Wheel and Axle,
235 of the Screw,
235 Miscellaneous Questions,
246 Habits of a man of business, 251 Forms of Notes, Bonds, &c., 251
EXPLANATION OF CHARACTERS USED IN
Equality is denoted by two horizontal lines. + Addition; as 4+3=7, which signifies that 4 added
to 3 equals 7. X Multiplication; as 4X3=12, which signifies that 4
multiplied by 3 equals 12. Subtraction; as 3–4=l, which signifies that 3 taken
from 4 leaves 1. )(, ;, , 24, Division ; as 2)4(2 and 4; 2=2, and
32 and 2/4=2; in either case it is signified that A divided by 2 equals 2. : : Proportion; as 2 : 4 ::6 12; which is read, 2 is to 4 as 6 is to 12.