process hereafter described, if required: of which it must be observed, that the trouble is great only when the quantity of goods in question is great. The purchaser takes his test-weight (say, one pound) to the shop or warehouse with him. With this one pound he can test every one of the shopkeeper's weights, as follows. Suppose he wants to buy 71 pounds of goods. On his own card, or on that of the seller, he sees G, C, B, A, which are the weights he has a right to. He produces his own test-weight (which call T), and requires the seller to show him that If this be found true, and if only one of the preceding be tried twice, changing the scales the second time, then neither scales nor weights can be wrong. It must be observed, that by this method the proper officers might make the usual examination of weights without carrying more weights about with them than one pound, one ounce, and one grain. Secondly. Suppose it is allowed to place weights in both scales. Then, it is sufficient if each weight (the first being one pound) be three times as great as the other. That, is, any number of pounds under 122 may be weighed with weights of A table might be formed as before. It is impossible to carry this principle further without introducing two weights of a sort. The two mathematical theorems on which the preceding rests are as follows: 1. Any number whatever may be made by adding together single terms out of the series 2. Any number may be made by adding and subtracting single terms out of the series 1 3 9 27 81 243 729 &c. The first of the preceding theorems explains a common puzzle, in which a number of cards is handed to the person to be mystified, and he is desired to think of a number, and tell on which cards it is to be found. The person in the secret then tells the number. By putting down the cards for all numbers up to 31 inclusive, as follows, the reader will easily find out the trick. II. There is a very large class of amusing properties of numbers, which the greater part of readers never know, not because there is anything difficult in announcing them, but simply because a few mathematical terms stand in the way. To prove these theorems is a work of study; but to understand them is not. We shall in this article explain some terms, and then exhibit the use which can be made of them in abbreviating propositions which in common language would lose all their interest, on account of the cloud of words in which they would be wrapped. The reader will see whether he understands each term by looking at the questions which follow it. 1. Multiple. 30 is called a multiple of 10, because it contains an exact number of tens, namely 3 tens. It is also a multiple of 6, 5, 15, 2, 3, and 1 It is not, properly speaking, a multiple of itself; but it is usual to say that 30 is the first multiple of 30, as it contains itself once exactly. Thus the multiples of 3, are 369 12 15 18 21 &c. Question. Why is every multiple of 10 also a mulple of 5 ? 2. Prime. A prime number is one which is a multiple of no number but 1 (which all numbers are) and itself. Thus 13 is an exact number of ones, and of thirteens, but of nothing else: it is then what is called a prime number. The series of prime numbers, up to 41, is as follows: 1 2 3 5 7 11 13 17 19 23 29 31 37 41. Question. Why is it that a prime number (excepting 2 and 5) cannot end with 0, 2, 4, 5, 6, or 8? 3. Square. A square number, or a second power, is a name given to a number which is made by multiplying a number by itself. Thus 9 is a square number (3 times 3); 10 is not. The series of square numbers is 149 16 25 36 49 64 81 100 &c. Questions. Why is it that a square number cannot end with 2, 3, or 7? Why is it that the square of an even number must be even, and of an odd number odd? 4. Cube. A cube number, or a third power, is a name given to a number which is made by multiplying a number twice by itself, or multiplying three equal numbers together; such as 64 (4 times 4 times 4; or 4, 4, and 4 multiplied together). The series of cubes is as follows:1 8 27 64 125 216 343 512 729 1000 &c. 5. Fourth powers, Fifth powers, &c. These are names given to numbers which arise from multiplying together four equal numbers, five equal numbers, &c. Thus, 2, 2, 2, 2, 2, multiplied together, give 32, which is the fifth power of 2. The eighth power of 3 is 6561. Questions. Why is it that every fourth power is a square; every sixth power a cube, and also a square? III. We shall now proceed to notice some relations between numbers, such as may easily be verified. 1. Odd numbers added together, beginning from 1, and not leaving out any, always give a square number. For instance, 1 and 3 1, 3, and 5 make 4, or the square of 2 9 16 3 4 and so on. If a person who knows this were to lay a bet with another that he would form the squares of all numbers under 100 sooner than the other, he would certainly win; for while the second would have to perform a multiplication at each step, the first would proceed as follows: : 2. An uninterrupted sum of cubes, beginning from 1, is always a square number. The series of cubes is, 1 8 27 64 125 216 343 &c. 3. Every number is either the sum of two, three, or four square numbers. The square numbers are, 1 4 9 16 25 36 49 &c. and the following are instances of numbers in the first column, and the squares which compose them in the 4. Every odd number can be made up of three square numbers at most, except those which, when divided by 8, leave a remainder 7. In the first column following, are odd numbers; in the second, the remainders of the same odd numbers divided by 8; in the third, the least number of squares of which the said number may be composed, which is never necessarily four, except when the remainder is 7. |