Here any figure in the first place, reckoning from right to left, denotes only its own simple value; but that in the second place, denotes ten times its simple value; and that in the third place, a hundred times its simple value; and so on; the value of any figure, in each successive place, being always ten times its former value. Thus, in the number 1796, the 6 in the first place denotes only six units, or simply six; 9 in the second place signifies nine tens, or ninety; 7 in the third place, seven hundred; and the 1 in the fourth place, one thousand; so that the whole number is read thus, one thousand seven hundred and ninety-six. As to the cipher 0, it stands for nothing of itself, but being joined on the right hand side to other figures, it increases their value in the same tenfold proportion : thus, 5 signifies only five; but 50 denotes 5 tens, or fifty; and 500 is five hun dred; and so on. For the more easily reading of large numbers, they are divided into periods and half-periods, each half-period consisting of three figures; the name of the first period being units; of the second, millions; of the third, millions of millions, or bi-millions, contracted to billions; of the fourth, millions of millions of millions, or tri-millions, contracted to trillions; and so on. Also the first part of any period is so many units of it, and the latter part so many thousands. The following Table contains a summary of the whole doctrine: Periods. Quadrill. Trillions; Billions; Millions; th. un. 15080 72003 109026 483500 2500639 7523000 Units. AA 14 da d4d4 Figures. 123,456; 789,098; 765,432; 101,234; 567,890. EXAMPLES. Express in words the following numbers; viz. 34 96 180 304 6134 9028 th. un. NUMERATION is the reading of any number in words that is proposed or set down in figures, which will be easily done by the help of the following rule, deduced from the foregoing tablets and observations, viz. Divide the figures in the proposed number, as in the summary above, into periods and half-periods; then begin at the left-hand side, and read the figures with the names set to them in the two foregoing tables. 13405670 47050023 309025600 4723507689 274856390000 6578600307024 NOTATION is the setting down in figures any number proposed in words; which is done by setting down the figures instead of the words or names belonging to them in the summary above; supplying the vacant places with ciphers where any words do not occur. Set down in figures the following numbers: Fifty-seven. Two hundred and eighty-six. Nine thousand, two hundred and ten. 1 = I. 2 = II. 3 = III. 4 Twenty-seven thousand, five hundred and ninety-four. Six hundred and forty thousand, four hundred and eighty-one. Twenty-seven thousand and eight millions, ninety-six thousand, two hundred and four. Two hundred thousand and five hundred and fifty millions, one hundred and ten thousand, and sixteen. Twenty-one billions, eight hundred and ten millions, sixty-four thousand, one hundred and fifty. OF THE ROMAN NOTATION. THE Romans, like several other nations, expressed their numbers by certain letters of the alphabet. The Romans used only seven numeral letters, being the seven following capitals: viz. I for one; V for five; X for ten; L for fifty; C for a hundred; D for five hundred; M for a thousand. The other numbers they expressed by various repetitions and combinations of these, after the following manner: 5 6 = VI. 7 = VII. 8 = VIII. 9 = IX. 100 500 IIII or IV. 10 = X. 50 L. = C. D or I). 1000 2000 = 5000 V or 105. = 2000000 &c. EXAMPLES. M or CIO. 10000 or CCI. 50000 Lor 1999. 100000 Cor CCC. 1000000 &c. As often as any character is repeated, so many times is its value repeated. A less character before a greater diminishes its value. A less character after a greater increases its value. M or CCCCIƆƆ20. MM. For every annexed, this becomes ten times as many. For every C and Ɔ, placed one at each end, it becomes ten times as much. A bar over any number, increases it 1000 fold. EXPLANATION OF CERTAIN CHARACTERS. THERE are various characters or marks used in Arithmetic and Algebra, to denote several of the operations and propositions; the chief of which are as follow: + signifies plus, or addition. X = :::: = ... ...... ... ......... ... ... ... ... ...... ... ......... minus, or subtraction. multiplication. division. proportion. ✓ V Thus, 5+3, denotes that 3 is to be added to 5. 6 - 2, denotes that 2 is to be taken from 6. 7 × 3, denotes that 7 is to be multiplied by 3. 84, denotes that 8 is to be divided by 4. 2:34:6, shows that 2 is to 3 as 4 is to 6. 6410, shows that the sum of 6 and 4 is equal to 10. √3, or 31, denotes the square root of the number 3. square root. cube root, &c. 3/5, or 5, denotes the cube root of the number 5. 72, denotes that the number 7 is to be squared. 83, denotes that the number 8 is to be cubed. &c OF ADDITION. ADDITION is the collecting or putting of several numbers together, in order to find their sum, or the total amount of the whole. This is done as follows: Set or place the numbers under each other, so that each figure may stand exactly under the figures of the same value; that is, units under units, tens under tens, hundreds under hundreds, &c; and draw a line under the lowest number, to separate the given numbers from their sum, when it is found. Then add up the figures in the column or row of units, and find how many tens are contained in their sum.-Set down exactly below, what remains more than those tens, or if nothing remains, a cipher, and carry as many ones to the next row as there are tens. Next add up the second row, together with the number carried, in the same manner as the first. And thus proceed till the whole is finished, setting down the total amount of the last row. TO PROVE ADDITION. First Method. Begin at the top, and add together all the rows of numbers downwards, in the same manner as they were before added upwards; then if the two sums agree, it may be presumed the work is right.-This method of proof is only doing the same work twice over, a little varied. Second Method.-Draw a line below the uppermost number, and suppose it cut off.—Then add all the rest of the numbers together in the usual way, and set their sum under the number that is to be proved. Lastly, add this last found number and the uppermost line together; then if their sum be the same as that found by the first addition, it may be presumed the work is right. This method of proof is founded on the plain axiom, that "The whole is equal to all its parts taken together," Third Method. Add the figures in the uppermost line together, and find how many nines are contained in their sum.-Reject those nines, and set down the remainder towards the right hand directly even with the figures in the line, as in the next example. Do the same with each of the proposed lines of numbers, setting all these excesses of nines in a column on the right hand, as here 5, 5, 6. Then, if the 18304 excess of 9's in this sum, found as before, be excess of 9's in the total sum 18304, the work is right. Thus, the sum of the right hand column 5, 5, 6, is 16, the excess of which above 9 is 7. Also the sum of the figures in the sum total 18304 is 16, the excess of which above 9 is also 7, the same as the former.* equal to the OTHER EXAMPLES. 2. 12345 3. 12345 67890 67890 43210 IT 9876 12345 67890 9 302445 90684 290100 78339 302445 90684 3197 6512 8295 4. 12345 EXAMPLE 1. 876 9087 56 234 1012 23610 11265 23610 Ex. 5. Add 3426; 9024; 5106; 8390; 1204 together. 6. Add 509267; 235809; 72910; 8392; 420; 21; and 9 together. Excess of nines Ans. 27150. Ans. 826828 7. Add 2; 19; 817; 4298; 50916; 730205; 9120634 together. Ans. 9906891. 8. How many days are in the twelve calendar months? Ans. 365. • This method of proof depends upon a property of the number 9, which, except the number 3, beongs to no other digit whatever; namely, "that any number divided by 9, will leave the same remainder as the sum of its figures or digits divided by 9;" which may be demonstrated in this manner. Demonstration-Let there be any number proposed, as 4658. This, separated into its several parts, becomes 4000 +690 +50+8. But 4000 4 X 1000 4 X (999 + 1) = 4 × 999 +4. In like manner 600)=6 × 99 +6; and 50 = 5 X 9+ 5. Therefore the given number 4658 = 4 × 999 + 4 + 6 × 99 +6+5×9+5+ 8 = 4 × 999 + 6 × 99 + 5 × 9 + 4 + 6 + 5 +8; and 4658÷9=(4 × 999 + 6 × 99+5×9+4+ 6 + 5 +8) ÷ 9. But 4 × 999 +6 × 99 +5 × 9 is evidently divisible by 9, without a remainder; therefore if the given number 4658 be divided by 9, it will leave the same remainder as ♦ + 6 + 5 + 8 divided by 9. And the same, it is evident, will hold for any other number whatever. In nike manner, the same property may be shown to belong to the number 3; but the preference is usually given to the number 9, on account of its being more convenient in practice. Now, from the demonstration above given, the reason of the rule itself is evident; for the excess of in two or more numbers being taken separately, and the excess of 9's taken also out of the sum of the former excesses, it is plain that this last excess must be equal to the excess of 9's contained in the total sum of all these numbers; all the parts taken together being equal to the whole.-This rule was first given by Dr Wallis in his Arithmetic, published in the year 1657 EXPLANATION OF CERTAIN CHARACTERS. THERE are various characters or marks used in Arithmetic and Algebra, to denote several of the operations and propositions; the chief of which are as follow : + signifies plus, or addition. square root. cube root, &c. 7 x 3, denotes that 7 is to be multiplied by 3. V5, or 5, denotes the cube root of the number 5. &c OF ADDITION. ADDITION is the collecting or putting of several find their sum, or the total amount of the whole. Set or place the numbers under each other, so that each figure may stand exactly under the figures of the same value; that is, units under units, tens under tens, hundreds under hundreds, &c; and draw a line under the lowest number, to separate the given numbers from their sum, when it is found. Then add up the figures in the column or row of units, and find how many tens are contained in their sum.-Set down exactly below, what remains more than those tens, or if nothing remains, a cipher, and carry as many ones to the next row as there are tens.—Next add up the second row, together with the number carried, in the same manner as the first. And thus proceed till the whole is finished, setting down the total amount of the last row. numbers together, in order to This is done as follows: TO PROVE ADDITION. First Method. Begin at the top, and add together all the rows of numbers downwards, in the same manner as they were before added upwards; then if the two sums agree, it may be presumed the work is right.—This method of proof is only doing the same work twice over, a little varied. Second Method.-Draw a line below the uppermost number, and suppose it cut off. Then add all the rest of the numbers together in the usual way, and |