Here any figure in the first place, reckoning from right to left, denotes its own simple value; but that in the second place, denotes ten times its si value; and that in the third place, a hundred times its simple value; a on; the value of any figure, in each successive place, being always ten time former value. Thus, in the number 1796, the 6 in the first place denotes only six uni simply six; 9 in the second place signifies nine tens, or ninety; 7 in the place, seven hundred; and the 1 in the fourth place, one thousand; so tha 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 hand side to other figures, it increases their value in the same tenfold propo thus, 5 signifies only five; but 50 denotes 5 tens, or fifty; and 500 is five dred; and so on. For the more easily reading of large numbers, they are divided into perio half-periods, each half-period consisting of three figures; the name of th period being units; of the second, millions; of the third, millions of milli bi-millions, contracted to billions; of the fourth, millions of millions of milli tri-millions, contracted to trillions; and so on. Also the first part of any 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; Half-per. thun. th. un. th. un. thun. Units. th. un. Figures. 123,456; 789,098; 765,432; 101,234; 567,890. NUMERATION is the reading of any number in words that is proposed down in figures, which will be easily done by the help of the following r duced from the foregoing tablets and observations, viz. Divide the figures in the proposed number, as in the summary abo periods and half-periods; then begin at the left-hand side, and read the with the names set to them in the two foregoing tables. NOTATION is the setting down in figures any number proposed in which is done by setting down the figures instead of the words or names ing to them in the summary above; supplying the vacant places with where any words do not occur. EXAMPLES, Set down in figures the following numbers: Fifty-seven. Two hundred and eighty-six. Nine thousand, two hundred and ten. Twenty-seven thousand, five hundred and ninety-four. Six hundred and forty thousand, four hundred and eighty-one. Three millions, two hundred and sixty thousand, one hundred and six. Four hundred and eight millions, two hundred and fifty-five thousand, one hun dred and ninety-two. 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 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 an follow: minus, or subtraction. multiplication. division. :...... proportion. ... ... ... ...... ... ......... equality. square root. cube root, &c. 5 + 3, denotes that 3 is to be added to 5. 6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. 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, aud 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," EXAMPLE 1. 3197 6512 8295 Excess of nines 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. 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.* Then, if the 18304 equal to the Ex. 5. Add 3426; 9024; 5106; 8390; 1204 together. Ans. 27150. 6. Add 509267; 235809; 72910; 8392; 420; 21; and 9 together. Ans. 826828 7. Add 2; 19; 817; 4298; 50916; 730205; 9120634 together. Ans. 99 )689 1. 8. How many days are in the twelve calendar months? Aus. 365. * This method of proof depends upon a property of the number 9, which, except the number 3, beougs 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 + 600 + 50 + 8. But 4000 4 X 1000 = 4 × (999 + 1) = 4 × 999 +4. In like manner 600)=6 × 99 + 6, and 50 = 5 × 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 × +5×9+4+ 6 + 5 + 6) ÷ 9. But 4 × 999 +6 × 99 +5 × 9 is evidently divisible by 9, without ● remainder; therefore if the given number 4658 be divided by 9, it will leave the same remainder as 4+6+5+8d viced by 9. And the same, it is evident, will hold for any other number whatever. In like 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 B's 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.-1 his rule was first given by Dr Wallis in his Arithmetic, published in the year 1657 9. How many days are there from the 15th day of April to the 24th d November, both days included? Ans. 10. An army consisting of 52714 infantry* or foot, 5110 horse, 6250 drag 3927 light-horse, 928 artillery or gunners, 1410 pioneers, 250 sappers, and miners; what is the whole number of men? Ans. 7 OF SUBTRACTION. SUBTRACTION teaches to find how much one number exceeds another, their difference or the remainder, by taking the less from the greater. method of doing which is as follows : Place the less number under the greater, in the same manner as in Ad that is, units under units, tens under tens, and so on; and draw a line them. Begin at the right hand, and take each figure in the lower line or ber from the figure above it, setting down the remainder below it.-But figure in the lower line be greater than that above it, first borrow or add the upper one, and then take the lower figure from that sum, setting do remainder, and carrying 1, for what was borrowed, to the next lower figur which proceed as before, and so on till the whole is finished. TO PROVE SUBTRACTION. ADD the remainder to the less number, or that which is just above it, the sum be equal to the greater or uppermost number, the work is right. 574371. Ans. 7929231. 6. From 8503602 Take 7. Sir Isaac Newton was born in the year 1642, and he died in 1727; 1 was he at the time of his decease? Ans. 8 8. Homer was born 2573 years ago, and Christ 1840 years ago; then h before Christ was the birth of Homer? Ans. 73 • The whole body of foot soldiers is denoted by the word Infantry; and all those that c horseback, by the word Cavalry.-Some authors conjecture, that the term infantry is deri a certain Infanta of Spain, who, finding that the army commanded by the king her father defeated by the Moors, assembled a body of the people together on foot, with which she and totally routed the enemy. In honour of this event, and to distinguish the foot sold were not before held in much estimation, they received the name of Infantry, from her of Infanta. The reason of this method of proof is evident: for if the difference of two numbers to the less, it must manifestly make up a sur «qual to the greater, |