method should be pursued in every other instance, and the product will always be found in the line of the multiplier and under the multiplicand. 27. If we seek in the table of Pythagoras the product of 5 by 7, we shall find, as before, 35, although in this case 5 is the multiplicand, and 7 the multiplier. This remark is applicable to each product in the table, and it is possible, in any multiplication, to reverse the order of the factors; that is, to make the multiplicand the multiplier, and the multiplier the multiplicand. As the table of Pythagoras contains but a limited number of products, it would not be sufficient to verify the above conclusion by this table; for a doubt might arise respecting it in the case of greater products, the number of which is unlimited; there is but one method independent of the particular value of the multiplicand and multiplier of showing that there is no exception to this remark. This is one well calculated for the purpose, as it gives a good illustration of the manner, in which the product of two numbers is formed. To make it more easily understood, we will apply it first to the factors 5 and 3. If we write the figure 1, 5 times on one line, and place two similar lines underneath the first, in this manner, the whole number of 1s will consist of as many times 5 as there are lines, that is, 3 times 5; but, by the disposition of these lines, the figures are ranged in columns, containing 3 each. Counting them in this manner, we find as many times 3 units as there are columns, or 5 times 3 units, and as the product does not depend on the manner of counting, it follows that 3 times 5 and 5 times S give the same product. It is easy to extend this reasoning to any numbers, if we conceive each line to contain as many units as there are in the multiplicand, and the number of lines, placed one under the other, to be equal to the multiplier. In counting the product by lines, it arises from the multiplicand repeated as many times as there are units in the multiplier; but the assemblage of figures written presents as many columns as there Arith. 3 are units in a line, and each column contains as many units as there are lines; if, then, we choose to count by columns, the number of lines, or the multiplier, will be repeated as many times as there are units in a line, that is, in the multiplicand. We may therefore, in finding the product of any two numbers, take either of them at pleasure, for the multiplier. 28. The reasoning, just given to prove the truth of the preceding proposition, is the demonstration of it, and it may be remarked, that the essential distinction of pure mathematics is, that no proposition, or process, is admitted, which is not the necessary consequence of the primary notions, on which it is founded, or the truth of which is not generally established by reasoning independent of particular examples, which can never constitute a proof, but serve only to facilitate the reader's understanding the reasoning, or the practice of the rules. 29. Knowing all the products given by the nine first numbers, combined with each other, we can, according to the remark in article 23, multiply any number by a number consisting of a single figure, by forming successively the product of each order of units in the multiplicand, by the multiplier; the work is as follows; 526 3682 The product of the units of the multiplicand, 6, by the multiplier, 7, being 42, we write down only the 2 units, reserving the 4 tens to be joined with those, that will be found in the next higher place. The product of the tens of the multiplicand, 2, by the multiplier, 7, is 14, and adding the 4 tens we reserved, we make it 18, of which number we write only the units, and reserve the ten for the next operation. The product of the hundreds of the multiplicand, 5, by the multiplier, 7, is 35; when increased by the 1 we reserved, it becomes 36, the whole of which is written, because there are no more figures in the multiplicand. 30. This process may be given thus; To multiply a number of several figures by a single figure, place the multiplier under the units of the multiplicand, and draw a line beneath, to separate them from the product. Beginning at the right, multiplý successively, by the multiplier, the units of each order in the multiplicand, and write the whole product of each, when it does not exceed 9; but, if it contains tens, reserve them to be added to the next product. Continue thus to the last figure of the multiplicand, on the left, the whole result of which must be written down. Examples. 243 by 6. Ans. 1458. 8943 by 9. Ans. 80487. It is evident that, when the multiplicand is terminated by 0, the operation can commence only with its first significant figure; but to give the product its proper value, it is necessary to put, on the right of it, as many Os as there are in the multiplicand. As for the Os, which may occur between the figures of the multiplicand, they give no product, and a 0 must be written down when no number has been reserved from the preceding product, as is shown by the following examples: 360500 by 6. Ans. 2163000. 9097030 by 9. Ans. 81873270. 20508 by 5. Ans. 102540. 297000 by 7. Ans. 2079000. 31. The most simple number, expressed by several figures, being 10, 100, 1000, &c. it seems necessary to inquire how we can multiply any number by one of these. Now if we recollect the principle mentioned in article 6, by which the same figure is increased in value 10 times, by every remove towards the left, we shall soon perceive, that to multiply any number by 10, we must make each of its orders of units ten times greater; that is, we must change its units into tens, its tens into hundreds, and so on, and that this is effected by placing a 0 on the right of the number proposed, because then all its significant figures will be advanced one place towards the left. For the same reason, to multiply any number by 100, we should place two ciphers on the right; for, since it becomes ten times greater by the first cipher, the second will make it ten times greater still, and consequently it will be 10 times 10, or 100 times, greater than it was at first. Continuing this reasoning, it will be perceived that, according to our system of numeration, a number is multiplied by 10, 100, 1000, &c. by writing on the right of the multiplicand as many ciphers as there are on the right of the unit in the multiplier. 32. When the significant figure of the multiplier differs from unity, as, for instance, when it is required to multiply by SO, or 300, or 3000, which are only 10 times 3, or 100 times 3, or 1000 times 3, &c. the operation is made to consist of two parts, we at first multiply by the significant figure, 3, according to the rule in article 30, and then multiply the product by 10, 100, or 1000, &c. (as was stated in the preceding article) by writing one, two, three, &c. ciphers on the right of this product. Let it be required, for instance, to multiply 764 by 300. 764 229200 The four significant figures of this product result from the multiplication of 764 by 3, and are placed two places towards the left to admit the two ciphers, which terminate the multiplier. In general, when the multiplier is terminated by a number of ciphers, first multiply the multiplicand by the significant figure of the multiplier, and place, after the product, as many ciphers as there are in the multiplier. Multiply Examples. 35012 by 100. Ans. 3501200. 638427 by 500. Ans. 319213500. 2107900 by 70. Ans. 147553000. 9120400 by 90. Ans. 820836000. 33. The preceding rules apply to the case, in which the multiplier is any number whatever, by considering separately each of the collections of units of which it is composed. To multiply, for instance, 793 by 345, or, which is the same thing, to repeat 793, 345 times, is to take 793, 5 times, added to 40 times, added to s00 times, and the operation to be performed is resolved into 3 others, in each of which the multipliers, 5, 40, and 300, have but one significant figure. To add the result of these three operations easily, the calculation is disposed thus ; 793 345 3965 31720 237900 273585 The multiplicand is multiplied successively by the units, tens, hundreds, &c. of the multiplier, observing to place a cipher on the right of the partial product, given by the tens in the multiplier, and two on the right of the product given by hundreds, which advances the first of these products one place towards the left, and the second, two. The three partial products are then added together, to obtain the total product of the given numbers. As the ciphers, placed at the end of these partial products, are of no value in the addition, we may dispense with writing them, provided we take care to put in its proper place the first figure of the product given by each significant figure of the multiplier; that is, to put in the place of tens the first figure of the product given by the tens in the multiplier; in the place of hundreds the first figure of the product given by the hundreds in the multiplier, and so on. s4. According to what has been said, the rule is as follows. To multiply any two numbers, one by the other, form successively (according to the rule in article 30,) the products of the multiplicand, by the different orders of units in the multiplier; observing to place the first figure of each partial product under the units of the same order with the figure of the multiplier, by which the product is given; and then add together all the partial products. 35. When the multiplicand is terminated by ciphers, they may at first be neglected, and all the partial multiplications begin with the first significant figure of the multiplicand; but after |