- = Equality Explanation of Characters. is expressed by two horizontal marks; thus, 100 cts. 1 dollar, signifies that 100 cents are equal to 1 dollar. is expressed by a cross formed by one horizontal, and +Addition one perpendicular mark. Thus 4+9=9 signifies that 5 added to 4, is equal to 9. is expressed by one horizontal mark between the -Subtraction numbers. Thus 7-4-3 signifies that 4 taken from 7, the remainder is equal to 3. { is expressed by a cross formed by two oblique Multiplication marks. Thus 5×3-15 signifies that 5 multiplied by 3, the product is equal to 15. →or) (Division is expressed by a horizontal line with a dot on each side, or by a reversed parenthesis. Thus 6÷2=3, or 2)6(5 signifies that 6 divided by 2, the quotient is equal to 3. :::: Proportion has to 24. 8 is expressed by four colons. Thus 2:6: 8:24 signifies that 2 has the same proportion to 6, that 8 In arithmetical proportion two of the colons are placed horizontally; thus 2.4 :: 6. 8. 2 signifies the second power, or square of the number over which it is placed. Thus 62 denotes the square of 6, or 6×6=36. signifies the third power, or cube of the number. Thus 6 denotes the cube of 6, or 6×6×6=216. signifies the square root. signifies the cube root. Thus 364 denotes the square root of +2=2, shows that the sum of 2 and 4, subtracted from 8, is equal to 2. The line over the 4 and 2 is called a vinculum. INTRODUCTION. ARITHMETICK. Arithmetick is the science of numbers, and is of two kinds, theoretick and practical. Theoretick Arithmetick treats of the nature of numbers, and shews the foundation of the rules for practical operations.* Practical Arithmetick is the method of applying these rules to the solution of questions and the transaction of business. In entering upon the study of Arithmetick, the first objects which demand the student's attention are Notation and Numeration. With these he should endeavour to become familiar, as a knowledge of them is indispensable at every step of his future progress. NOTATION. Notation is the method of writing, or expressing, any proposed number in characters or figures. There are two methods of expressing numbers, the Roman and the Arabick. The Roman method is by letters of the alphabet. It is at present but little used, except in numbering chapters, sections, and the like. The Arabick method is by characters, and is the one in general use. The Arabick characters or figures are the ten following; 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, and 0 cipher.† By repeating and varying the position of these ten characters, all numbers whatever may be expressed. The nine first are called significant figures, or digits, because they invariably point out, or express some particular number. The cipher standing alone has no signification; but placed at the right hand of a siguificant figure, it increases the value of that figure in a ten fold proportion. Thus 4 standing alone is four; annex a cipher (40,) and it is forty; another cipher (400,) and it is four hundred. Hence figures have a local as well as simple value ; and the local value depends on the distance of the figure from the right hand, or place of units, each removal to the left increasing the value of the figure in a ten fold proportion. And the effect is the same, whether the places on the right be filled with ciphers or significant figures. But this will appear more clearly from what follows. *The Theory of Arithmetick contained in this treatise will be mostly found in the Notes. DC. + These characters were introduced into Europe, from Arabia, by the Saracens about A.D. 991. The letters of the alphabet were used previous to this time. The following table exhibits a comparison of the two methods of notation. 1 I. 6 VI. 11 XI.16 XVI.110 X: 60 LX. 100 2 II. 7 VII. 12 XII. 17 XVII. 20 XX. 70 3III. 8VIII. 13XIII. 18XVIII. 30XXX. 4IV. 9 IX. 14XIV. 19 XIX. 40 XL. 90 5 V. 10 X. 15 XV. 20 XX.50 L. 100 LXX. 200 80LXXX. 300 C. 600 DCC. XC.400 CCCC. 900 DCCCC. M. NUMERATION. Numeration teaches how to read any proposed number expressed in figures. We have already observed that figures have a local as well as simple value. The method of determining these values may be learnt from the following The words at the head of the Table show the names of the several places against which they stand, or the local value of the figures occupying those places. These words should, in the first place, be perfectly committed to memory, as they are applicable to any sum or number. This being done, to enumerate any sum, observe the following RULE To the simple value of each figure, join the name of its place, beginning at the left, and reading towards the right. Thus 8 being in the ten's place in the table, is eighty, 9 in the hundred's place is nine hundred, 7 in the thousand's place is seven thousand, &c. Hence the whole number would read thus: six trillions, four hundred and sixty-eight thousands, seven hundred and sixtyfour billions, two hundred and thirteen thousands eight hundred fifty-one. millions, two hundred sixty-seven thousands nine hundred and eighty-seven. Application. Eight. Nine Write the following numbers in their proper figures. teen. Eighty. Three hundred and sixty-nine. Five thousand, three hundred and seven. Thirty thousand and fifty-nine. Two billions. Three triilions, six billions, seven millions and one hundred thousands. Write the following numbers in words: 9, 26, 348, 4090, 84704, 514242, 42357440, 800000, 6000000, and 260400100220160. For the more easy reading of large numbers, it is customary to divide them into periods and half periods, as in the following Table : Periods. Sex. Quintili. Quadril. Trillions. Billions. Millions. Units. Half fer. th. un. th. un. th. un. th. un. th. un. th. un. cxt. cxu. Figures. 211.974 321,234 108,642 320,123 458.620 512.345 254,162 Here it will be observed that in enumerating, the same names are repeated in each of the periods, and then the name of the period annexed. Thus the first period is two hundred fifty-four thousand, one hundred and sixty-two units; the second, five hundred, twelve thousand, three hundred forty-five millions; the third, four hundred, fifty-eight thousand, six hundred and twenty billions, &c. Hence a number consisting of twenty, thirty. forty, or more figures, after dividing it into periods, and knowing the name of each, can be enumerated with the same ease as one consisting of six figures only. PART I. FUNDAMENTAL RULES. THE Fundamental Rules of Arithmetick are four, Addition, Subtraction, Multiplication and Division. They are called Principal or Fundamental Rules, because one, or more of them is concerned in all arithmetical operations. Each of these rules is of two kinds, simple and compound. They are simple when the numbers employed are all of one sort, or denomination; and compound when the numbers employed are of different denominations. After having made himself familiar with Notation and Numeration, the scholar's next business is to obtain a thorough knowledge of the four fundamental rules. If these are passed over slightly, he will, in his future progress, meet with continual embarrassment. But if he becomes master of each rule before he proceeds to the next, those difficulties, which would otherwise obstruct his progress, will entirely vanish, or be surmounted with ease. SECTION I. SIMPLE RULES. 1. Simple Addition. SIMPLE ADDITION is the putting together of several numbers of the same denomination, into one whole or total number, called the sum, or amount. Thus 5, 4 and 3 put together, their sum is 12. RULE.*—1. Write the numbers to be added under one another with units under units, tens under tens, and so on, beginning at the * This rule and the method of proof, are founded on the well known axiom, "the whole is equal to the sum of all its parts." The reason of carrying for ten is that ten in any inferior column, is just equal to one in the next superior, as is evident from the nature of Notation. There are several other methods of proving Addition, besides those given above. A very ingenious one is by casting out the nines, and as this method, with the proper variations, is applicable to all the fundamental rules, I shall proceed to explain it. The nines are cast out of a sum by beginning at one end of the line of figures, and adding them together, rejecting nine from the sum as often as it occurs. Example.-Cast the nines out of 14838. Beginning at the right hand, say 8 to 3 is 11, this being 9 and 2 over, drop the 9 and say 2 to 8 is 10, reject again, and say 1 to 4 is 5, and 1 is 6. 6 then is the excess after casting the 9's out of 14838. Proceed in the |