For the better understanding of the principles of Notation here explained, the following Table is given. Hundreds of Millions Tens of Millions Millions Hundreds of Thousands Hundreds Thousands Tens Units 4321 5 4 3 2 1 Three hundred and twenty-one. Four thousand 321. Fifty-four thousand 321. 6 5 4 3 2 1 654 thousand 321. 6 5 4 3 2 1 Seven millions 654 thous. 321. 8 7 6 5 4 3 2 1 Eighty-seven millions 654, 32k. 9 8 7 6 5 4 3 2 1 987 millions, 654, 321. The first column on the right hand contains units, and the figure 1 in that column, represents the number One. The second line consists of 1 Unit and two Tens, or twentyone; the third line of 3 hundreds, 2 tens, and 1 unit, or three hundred and twenty-one; and in the same manner the lowest line contains 9 hundreds of millions, & tens of millions, and 7 millions; in all 987 millions; also 6 hundreds of thousands, 5 tens of thousands, and 4 thousands ; in all 654 thousands; and lastly, 3 hundreds, 2 tens, and I unit: so that the whole sum expressed by the 9 figures in the lowest line, is nine hundred and eighty-seven millions, six hundred and fifty-four thousands, three hundred and twenty-one. In the same manner, Numeration may be carried on to any extent, as in the following example. Where 19 figures represent the sum Three Trillions, two hundred and twelve thousand, three hundred and fortyfive Billions, six hundred and seventy-eight thousand, nine hundred and eighty-seven Millions, six hundred and fiftyfour Thousands, three hundred and twenty-one. OF ADDITION. THE fundamental operations of Arithmetical calculation are these four; Addition, Subtraction, Multiplication, and VOL. I. Division: Division or rather, as quantities are susceptible of no other modification but augmentation and diminution; the two last operations, Multiplication and Division, are in fact only speedy methods of performing the two first operations, Addition and Subtraction. 9 7 5 3 Sum 24 By Addition we assemble and express on paper, the aggregate value of a number of separate quantities. When the quantities or the numbers by which they are expressed, consist of only one place of figures, as when 3, 5, 7, and 9, are to be added together, we say thus; three and five are eight, and seven are fifteen, and nine are twenty-four, writing 2 for the number of tens, and 4 for the remaining units, as in the margin: But when the sums to be added together consist of more than one place of figures, the scholar must be careful to place them so as that Units shall be immediately under units, Tens under tens, Hundreds under hundreds, &c. as in the anannexed example, where the inhabitants of the principal towns of a certain county, being calculated to be 4,386, 2,285, 7,309, 3,025, and 1766; it is required to know the amount of the population of these five towns.. Thousands. Hundreds. Tens. Units. 4386 2285 7309 3025 1766 18771 Write down these several sums, as in the margin; then drawing a line under the whole, say 6 and 5 are 11, and 9 are 20, and 5 are 25, and 6 are 31; that is, 3 tens and 1 unit; then write this in the place of units, and carrying (as it is termed) the three tens to the second column of figures, say, 3 and 6 are 9, and 2 are 11, and (passing over the nought) 8 are 19, and 8 are 27: here are 2 tens and 7 units, which units are to be written under the second column, and the 2 tens carried or added to the third column. Then say 2 and 7 are 9, and 3 are 12, and 2 are 14, and 3 are 17; where the 7 units are to be written under the column now summed summed up, and the ten is to be carried to the fourth column; saying, 1 and 1 are 2, and 3 are 5, and 7 are 12, and 2 are 14, and 4 are 18. This being the last column, the sunits are written under the figures added together, and the ten comes to occupy an additional place to the left hand hence we find the whole amount of the population of the five towns to be eighteen thousand seven hundred and seventy-one persons. It is of the utmost importance in business to be able to perform Addition with dispatch and accuracy; the learner ought therefore to practise it repeatedly, with sums of various lengths; and if he can readily add two simple units (which are also called Digits) together, he will easily add a Digit to a higher number: thus, 6 and 9 are 15, and 36 and 9 are 45. In summing up a long column of figures, where mistakes may happen, from interruption or other accidents, it is proper to write down the full amount of each column, either on a separate paper, or in the way shown in the 7652 .428 56865 .317 3509 ...31 ..14. .26.. 21... 5.... 73771 margin; by which means, should any error be suspected, each column of figures may be examined separately, without its being necessary to repeat the whole operation. Here the 1st. column amounts to 31, the 2d. to 14, of which the 4 is placed under the 5 tens of the 1st column: and so on with the others; and by the last addition of these seve ral amounts, the total 73,771 is obtained, in the same way as if the several numbers of tens had been carried to the succeeding column, as before directed. To ascertain the accuracy of Addition, several methods. have been devised, as the following: 1st. to repeat the operation, beginning at the top of the column, and adding the figures downwards: ed. to divide the column, if it be long, into several portions, and add each separately; the total of these parts added together, ought to be equal to the total of the whole column taken at once: 3d. to eut off a line, the uppermost for instance, of the account, and then add the remaining lines, the amount of which added to the line cut off, should be the same with the total first found, thus, Total of this last sum and 1st line 5488492 Whatever be the quantity adopted for the unit, it may be supposed to be divided into a number of equal parts; and these parts may be of any determined magnitude: but if, for example we should say, that the Pound of Sterling money is divided into 960 Farthings, it would be found extremely difficult either to reckon or to form a distinct conception of such a number of individual farthings, or of intermediate sums between 1 and 960. For this reason the Pound is first divided into 20 equal parts called Shillings; each shilling into 12 equal parts, called Pence; and each penny into 4 equal parts called Farthings; so that 1 Pound will contain 20 Shillings, or 240 Pence, or 960 Farthings. When a sum is given consisting of one or more units, together with one or more of these. subdivisional parts, it is said to be a complex sum; as 25 Pounds, 14 Shillings, 9 Pence, 3 Farthings; or written in this manner, £. 25 14s. .. 9d... 3 qrs. where the mark L. stands for the Latin term Libra, a Pound in weight, such a quantity of Silver having originally been the value of a Pound Sterling; Sh, for Shillings: D. being the first letter of the Latin word Denarius, a Denier or Penny; and Qrs. for Quadrans |