the sum total 18304, is 16, the excess of which above 9 is also 7, the same as the former*. Ex. 5. Add 3426; 9024; 5106; 8890; 1204, together. Ans. 27650. 6. Add 509267; 235809; 72920; 8392; 420; 21; and 9, together. Ans. 826838. 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 are digits divided by 9:" which may be demonstrated in this manner. 8 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 = (59) +5. Therefore the gi ven number 4658 = (4 × 999) + 4+ (6 × 99)+6+(5×9)+5+ (4 × 999) + (6 × 99) + (5 × 9) + 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 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 9'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.-This rule was first given by Dr. Wallis in his Arithmetic, published in the year 1657. SUBTRACTION. 7. Add 2; 19; 817; 4298; 50916; 730205; 9180634, Ans. 9966891. together. 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 Ans. 224. the 24th day of November, both days included? 10. An army consisting 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 the upper one, and then take the lower figure from that sum, setting down the remainder, and carrying 1, for what was borrowed, to the next lower figure, with which proceed as before; and so on till the whole is finished. 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. TO PROVE SUBTRACTION. ADD the remainder to the less number, or that which is just above it; and if the sum be equal to the greater or uppermost number, the work is right*. Ans. 257888. Ans. 4254165. Ans. 7929131. 4. From 5331806 take 5073918. 5. From 7020974 take 2766809. 6. From 8503402 take 574271. 7. Sir Isaac Newton was born in the year 1642, and he died in 1727: how old was he at the time of his decease? Ans. 85 years. 8. Homer was born 2560 years ago, and Christ 1827 years ago: then how long before Christ was the birth of Homer? Ans. 733 years. 9. Noah's flood happened about the year of the world 1656, and the birth of Christ about the year 4000: then how long was the flood before Christ? Ans. 2344 years. 10. The Arabian or Indian method of notation was first known in England about the year 1150: then how long is it since to this present year 1827 ? Ans. 677 years. 11. Gunpowder was invented in the year 1330: how long was that before the invention of printing, which was in 1441? Ans. 111 years. 12. The mariner's compass was invented in Europe in the year 1302 how long was that before the discovery of America by Columbus, which happened in 1492? Ans. 190 years. *The reason of this method of proof is evident; for if the difference of two numbers be added to the less, it must manifestly make up a sum equal to the greater. OF MULTIPLICATION. MULTIPLICATION is a compendious method of Addition, teaching how to find the amount of any given number when repeated a certain number of times; as, 4 times 6, which is 24. The number to be multiplied, or repeated, is called the Multiplicand. The number you multiply by, or the number of repetitions, is the Multiplier.-And the number found, being the total amount, is called the Product.-Also, both the multiplier and multiplicand are, in general, named the Terms or Factors. Before proceeding to any operations in this rule, it is necessary to learn off very perfectly the following Table, of all the products of the first 12 numbers, commonly called the Multiplication Table, or sometimes Pythagoras's Table, from its inventor. MULTIPLICATION TABLE. 2 3 4 5 6 7 8 9 10 11 12 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99108 -10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99110121 132 12 24 36 48 60 72 84 96 108 120132 144| To multiply any Given Number by a Single Figure, or by any · Number not exceeding 12. * Set the multiplier under the units' figure or right-hand place, of the multiplicand, and draw a line below it.-Then, beginning at the right-hand, multiply every figure in this by the multiplier.-Count how many tens there are in the product of every single figure, and set down the remainder directly under the figure that is multiplied; and if nothing remains, set down a cipher.-Carry as many units or ones as there are tens counted, to the product of the next figures; and proceed in the same manner till the whole is finished. EXAMPLE. Multiply 9876543210 the Multiplicand. 19753086420 To multiply by a Number consisting of Several Figures. † Set the multiplier below the multiplicand, placing them as in Addition, namely, units under units, tens under tens, &c. drawing a line below it. -Multiply the whole of the multiplicand by each figure of the multiplier, as in the last article ; + After having found the product of the multiplicand by the first figure of the multiplier, as in the former case, the multiplier is supposed to be divided into parts, and the product is found for the second figure in the same manner: but as this figure stands in the place of tens, the product must be ten times its simple value; and therefore the first figure of this product must be set in the place of tens; or, which is the same thing, directly under the figure multiplying by. And proceeding |