which depends on the place they stand in when joined together; as in the following table: 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, though it signify nothing of itself, yet being joined on the right-hand side to other figures, it increases their value in the same ten-fold proportion: thus, 5 signifies only five; but 50 denotes 5 tens, or fifty; and 500 is five hundred; 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. Figures. 123,456; 789,098; 765,432; 101,234; 567,890; And the whole may be thus read: -One hundred and twenty-three thousand, four hundred and fifty-six quadrillions; seven hundred and eighty-nine thousand, and ninety-eight trillions; seven hundred and sixty-five thousand, four hundred and thirty-two billions; one hundred and one thousand, two hundred and thirty-four millions; five hundred and sixty-seven thousand, eight hundred and ninety. NUMERATION is the reading of any number in words that is proposed or set down in figures; which will be easily done by help of the following rule, deduced from the foregoing tables 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. 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. 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, and one hundred and six. hundred 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 manner: 1=1 2 = II 3 = III 4= 1111 or iv 5 = v 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. 6 = VI 7 = VII 8 = VIII 9 = Ix 10 = x EXPLANATION OF CERTAIN CHARACTERS. There are various characters or marks used in Arithmetic and Algebra, to denote several of their operations and propositions †; the chief of which are as follow: S square root. cube root, &c. difference between two numbers when it is either not known, or not necessary to state, which is the greater. Thus, 5+3, denotes that 3 is to be added to 5. 6-2, denotes that 2 is to be taken from 6. 2:3:4:6, expresses that 2 is to 3 as 4 is to 6. 6 + 4 = 10, shows that the sum of 6 and 4 is equal to 10. 72, denotes that the number 7 is to be squared. See, farther, the definitions in Algebra. * To those students whose taste leads them to inquire into the History of Arithmetic, reference is especially made to Professor Leslie's Philosophy of Arithmetic, to the Rev. George Peacock's Treatise on Arithmetic, in the Encyclopædia Metropolitana, to a paper by the celebrated Humboldt, read before the Royal Academy of Berlin, of which a translation is printed in the Journal of the Royal Institution, vol. xxix.; and to a paper in the Bath and Bristol Magazine for Oct. 1833 (No. viii.) by Mr. Davies. Other references will be found in those works which, for want of room, must be omitted here. + All such symbols as designate operations to be performed, are called symbols of operation, and those which designate quantities of any kind are called symbols of quantity. 1 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 that 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 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 annexed 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 excess of 9s in this sum, found EXAMPLE I. 3497 5 6512 5 8295 6 18304 Excess of nines. 7 as before, be equal to the excess of 9s in the total sum 18304, the work is probably 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.* * This method of proof depends on a property of the number 9, which, except the number 3, belongs 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 × 1000=4×(999+1) = (4 x 999) +4. In like manner 600 = (6 x 99) + 6; and 50 = (5 x 9) + 5. Therefore the given number 4658 = (4x999)+4+(6×99)+6+ (5×9) + 5 + 8 = (4 x 999)+(6× 99)+(5x9)+4+6+5+8; and 46589 = (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 4+6+5+8 divided by 9. And the same, it is evident, will hold for any other number whatever. In Ex. 5. Add 3426; 9024; 5106; 8890; 1204, together. Ans. 27650. 6. Add 509267; 235809;72920; 8392;420; 21; and 9, together. Ans. 826838. 7. Add 2; 19; 817; 4298; 50916; 730205; 9180634, together. Ans. 9966891. 8. How many days are in the twelve calendar months? Ans. 365. 9. How many days are there from the 15th day of April to the 24th day of November, both days included ? Ans. 224. 10. An army consists of 52714 infantry*, or foot, 5110 horse, 6250 dragoons, 3927 light-horse, 928 artillery, or gunners, 1410 pioneers, 250 sappers, and 406 miners: what is the whole number of men? Ans. 70995. OF SUBTRACTION. SUBTRACTION teaches to find how much one number exceeds another, called their difference, or the remainder, by taking the less from the greater. The method of doing which is as follows : Place the less number under the greater, in the same manner as in Addition, that is, units under units, tens under tens, and so on; and draw a line below them. Begin at the right hand, and take each figure in the lower line, or number, from the figure above it, setting down the remainder below it. - But if the figure in the lower line be greater than that above it, first borrow, or add, 10 to 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. A similar property belongs to the number 11. Now, from the demonstration above given, the reason of the rule itself is evident: for the excess of 9s in two or more numbers being taken separately, and the excess of 9s taken also out of the sum of the former excesses, it is plain that this last excess must be equal to the excess of 9s 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. * The whole body of foot soldiers is denoted by the word Infantry; and all those that charge on horseback by the word Cavalry. Some authors conjecture that the term infantry is derived from a certain Infanta of Spain, who, finding that the army commanded by the king her father had been defeated by the Moors, assembled a body of the people together on foot, with which she engaged and totally routed the enemy. In honour of this event, and to distinguish the foot soldiers, who were not before held in much estimation, they received the name of Infantry, from her own title of Infanta. |