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The following explanations and table, not being contained in the Written Arithmetic, are inserted here for the convenience of those who have not studied the Mental Arithmetic. =EQUALITY iš expressed by two horizontal marks; thus 100 cts.=

1 dollar, signifies that 100 cents are equal to one dollar. +ADDITION is denoted by a cross, formed by one horizontal and one

perpendicular line, placed between the number; as 4+5=9,

signifying that 4 added to 5 equals 9. X MULTIPLICATION is denoted by a cross, formed by two oblique lines placed between the numbers; as 5x 3=15, signifying that

5 multiplied by 3, or 3 times 5 are equal to 15. --SUBTRACTION is denoted by one horizontal mark, placed between

the numbers; as 7–4=3, signifying that 4 taken from 7 leave 3. , or ;Division is denoted three different ways; 1st. by the re. versed parenthesis; 2dly. by a horizontal line placed between the numbers with a dor on each side of it; and 3dly. by writing the number to be divided over the other in the form of a fraction; thus 2)613, and 6+2=3 and 1=3, all signify the same thing, namely that if 6 be divided by 2 the quotient is 3.


1 2 3 4 | 5 | 6 | 7 | 8 9 10 | 11 12
2 | 4 6 8 | 10 | 12 | 14 | 16 18

1 20 | 22 24
31 6 9 | 12 | 15 | 18

36 41 8 12 16 20 24 28 32 36 40 447 48 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 45 50 55 60 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 54 60 66 72 7 | 14 | 21 | 28 | 35 | 42 | 49 56 63 70 77 84 8 16 24 | 32 | 40 | 48 56 64 72 80 88 96 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 99 | 108 10 20 30 40 50 60 70 80 90 100 110 120 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 99 | 110 | 121 | 132 12 | 24 36 | 48 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144

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Entered according to Act of Congress, in the year one thousand eight hundred and twenty eight, by ZADOCK THOMPSON, in the Clerk's Office of the District of Vermont.

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Section Louis Karpinski

A D V E RTISEMENT. 10-30-1951

(To the fourth edition. )

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When the improved edition of this work was published, in 1828, it was intended that the Written Arithmetic, which forms the second and third parts should always be accompanied by the Mental Arith. metic embraced in the first part. Since that time it has, however, been thought best to transpose such tables from the Mental to the Written Arithmetic, as to render the latter complete without the former, in order to lessen the expense of the book to those who do not wish to study mental arithmetic, or who have studied some other treatise ; and, thus prepared, it is now presented to the public. No alteration has been made from the last edition in the arrangement of the rules, and the whole of the second part is presented, as before, on the inductive plan of Lacroix. The principles are first developed by the analysis of familiar examples, and the method of applying these principles to the solution of questions is then expressed in general terms, forming a Rule, which is still further il lustrated by a great variety of practical questions. The analysis is printed in small type, occupies but little space, and may be omitted by those who wish to use rules without understanding them.

Addition and Multiplication, both involving the same principles, are presented in connexion, and also Subtraction and Division. A knowledge of decimals being necessary to a good understanding of our Federal currency and this knowledge being easily acquired by such as have learned the notation of whole numbers, decimals and Federal money are introduced immediately after the first section on simple numbers. By acquainting the pupil thus early with dicimals, he will be likely to understand them better and to avail himself of the facilities they afford in the solution of qnestions and the transaction of business.

Reduction ascending and descending are arranged in parallel columns and the answers to the questions of one column are found in the corresponding questions of the other. Compound multiplica. tion and division are arranged in the same way, and only one general rule for each is given, which was thought better than to perplex the pupil with a multiplicity of cases.

Interest and other calculations by the hundred are all treated decimally, that method being most simple and conformable to the notation of our currency. The nature and principles of proportion are fully developed and the method of applying them to the solution of questions clearly shown.

The written arithmetic of fractions being, to young pupils, some. what difficult to be understood, is deferred till they are made fa. mlliar with the most important arithmetical operations performed with whole numbers and decimals. The nature of roots and powers has been more fully explained in the present edition, and several new diagrams introduced for their elucidation. Throughout the

second part, it has been our main object to familiarize the pupil
with the fundamental principles of the science, believing that
when these are well understood, he will find no difficulty in applying
them to the particular cases which may occur.

The third part is mostly practical, and composed of such rules
and other matters as we conceived would be interesting and useful
to the student and the man of business.



Simple Interest


Varieties in Interest




Commission and Insurance 63 SECT. II.-MENSURATION 132

Interest on Notes

64 Mensuration of Superficies 132

Compound Interest

67 Mensuration of Solids




Loss and Gain




Equation of Payments

70 Fall of Heavy Bodies



72 Pendulums


Single Rule of Three

74 Mechanical Powers


Double Rule of Three








Assessment of Taxes





88 Table of Cylindric Measure 149


89 Table of Square Timber Measure 150

Integers treated as Fractions 90

Table-Log Measure 153

Multiplication and Division of Fr. 91 Log Table-Board Measure 154

Multiplication by Fractions 92 SECT. VI.-BOOK KEEPING 155

Division by Fractions



Fractions changed to other Fr. 94





70. An individual thing taken as a standard of comparison, is called unity, a unit, or one.

71. Number is a collection of units, or ones.

72. Numbers are formed in the following manner; one and one more are called two, two and one, three, three and one, four, four and one, five, five and one, six, six and one, seven, seven and one, eight, eight and one, nine, nine and one, ten; and in this way we might go on to any extent, forming collections of units by the continual addition of one, and giving to each collection a different name. But it is evident, that, if this course were pursued, the names would soon become so numerous that it would be utterly impossible to remember them. Hence has arisen a method of combining a very few names, so as to give an almost infinite variety of distinct expressions. These names, with a few exceptions, are derived from the names of the nine first numbers, and from the names given to the collections of ten, a hundred, and a thousand units. The nine first numbers, whose names are given above, are called units, to distinguish them from the collections of tens, hundreds, &c. The collections of tens are named ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety.(6). The intermediate numbers are expressed by joining the names of the units with the names of the tens. To express one ten and four units, we say fourteen, to express two tens and five units, we say twenty-five, and others in like manner. The collections of ten tens, or hundreds, are expressed by placing before them the names of the units; as, one hundred, two hundred, and so on to nine hundred. The intermediate numbers are formed by joining to the hundreds the collections of tens and units.

Το express two hundred, four tens, and six units, we should


say, two hundred forty-six. The collections of ten hundreds are called thousands, which take their names from the collections of units, tens and hundreds, as, one thousand, two those sand, ten thousand, twenty thousand, one hundred thous sand, two hundred thousand, &c. The collections of ten hundred thousands are called millions, the collections of ten hundred millions are called billions, and so on to trillions, quadrillions, &c. and these are severally distinguished like the collections of thousands. The foregoing names, combined according to the method above stated, constitute the spoken numeration.

73. To save the trouble of writing large numbers in words, and to render computations more easy, characters, or symbols, have been invented, by which the written expression of numbers is very much abridged. The method of writing numbers in characters is called Notation. The two methods of notation, which have been most extensively used, are the Roman and the Arabic.* The Roman numerals are the seven following letters of the alphabet, I, V, X, L, C, D, M, which are now seldom used, except in numbering chapters, sections, and the like. The Arabic characters are those in common use. They are the ten following: O cipher, or zero, 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine. The above characters, taken one at a time, denote all the numbers from zero to nine inclusive, and are called simple units. To denote numbers larger than nine, two or more of these characters must be used. Ten is written 10, twenty 20, thirty 30, and so on to ninety, 90; and the intermediate numbers are expressed by writing the excesses of simple units in place of the cipher; thus for fourteen we write 14, for twentytwo, 22, &c.(13) Hence it will be seen that a figure in the second place denotes a number ten times greater than it does when standing alone, or in the first place. The first place at the right hand is therefore distinguished by the nanie of units place, and the second place, which contains units of a

*A comparison of the two methods of notation is exhibited in the following

TABLE. 1=1 110=X 1100=C 11000=M orCl] 10000=x orCCIɔɔ 2=II 20=XX 200=CC 1100=MC 3=III 130=XXX|300=CCC 1200=MCC 4=IV 40=XL 400=CCCC 1300=MCCC 100000=CCCrɔɔɔ 5=V 50=L 500=D orlɔ 1400=MCCCC 1000000=M 6=VI 60=LX 600=DC 1500=MD

2000000=MM 7=VII 70=LXX 700 - DCC 12000=MM !1929=MDCCCXXIX 8=VIII 80=LXXX 500=DCCC 5000=1ɔɔ or v b=IX 90=XC 900=noccc 6000=vi


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