APPENDIX. In this Appendix we shall, by using the Roman notation, number the different portions the same as the Articles in the body of the work, which they are designed to explain or enlarge upon. II. The particular form of many of the symbols employed to indicate the relations of numbers, as well as the different operations which are required to be performed upon them, are no doubt entirely arbitrary. But some of them are easily traced to their true origin, and a reason is readily seen why they were employed. Under this article we propose to make a few remarks on what may be properly called the philosophy of arithmetical symbols. Not that we presume to know positively the reasons for the particular form of all symbols, but it is thought a few suggestions and queries on this subject might not be uninteresting to the inquiring mind. The most natural method which could be devised for indicating that one quantity or number was greater than another is the symbol >, or <, so placed between the quantities or numbers compared as to open towards the greater, showing that in passing from the angle of this symbol to the opposite side, there is an increase. In this sense, this same symbol is used in music. In conformity with this use of the above symbol we have $199 cents; 45 cents <$1; &c. The symbol $, employed to denote dollars is no doubt derived from a combination of the letters U and S, which, taken together, form the abbreviation for United States. If a com pound letter be formed by uniting these letters, it will at once be seen how small a modification it is necessary to make in it so as to obtain the symbol $. And when we consider that this kind of currency, having a dollar for its unit, is peculiarly that of the United States, the explanation seems quite satisfactory. Having agreed to employ the symbol >, or <, to denote inequality, we may ask how must it be modified when placed between two equal quantities so as to become the symbol of equality? In this case, there is no reason why this symbol should be placed with its openings towards the one quantity rather than the other, since neither is larger than the other. It ought to open equally towards each; hence it must of necessity become =, which is the symbol of equality. The addition of one quantity to another might very concisely be represented by the union of two straight lines. But in what manner shall these two lines be united? When numbers are united by addition, the first may be said to be added to the second, or the second may be said to be added to the first, so that the two terms united have mutual relations with each other. So also when two numbers are united by multiplication, they are each factors, and have mutual relations. And moreover since the operations of multiplication are so nearly allied to those of addition, it would seem natural to represent multiplication also by the union of two lines so placed as to be mutually related. If two equal lines mutually bisect each other at right angles, it would form a symbol which might be appropriately used to denote either addition or multiplication. When these lines are so placed as that one may be horizontal and the other perpendicular, as +, it is the symbol of addition. When this symbol is so placed that the lines are oblique as X, the symbol denotes multiplication. Since the symbol +, denotes the union of two numbers by addition, if we take one away, or perform the operation of subtraction, this operation might be indicated by the symbol + after one of the lines, as for instance the vertical one, had been taken away, so as to give for the symbol of subtraction. The symbol, for division, was probably suggested by the idea of separating or dividing by means of a line passing through the quantity, leaving a portion on each side of the straight line, which portions are represented by the two dots placed on the two sides of this straight line. If we substitute numbers for those dots, considering the one above the line to correspond with the dividend, and the one below the line to correspond with the divisor, we shall thus have, in the case of 7 divided by 5, the expression, which is in strict accordance with the notation for a vulgar fraction, which shows that a vulgar fraction is but another method of denoting a division where the numerator corresponds with the dividend, and the denominator with the divisor. If in the proportion 6: 38: 4, we draw a straight horizontal line between the two dots separating the first and second terms, as well as between the two dots which separate the third and fourth terms; also draw two horizontal lines joining the four dots between the second and third terms, we shall have 6÷÷3:8÷4, which is read, 6 divided by 3 is equal to 8 divided by 4, and this is precisely what the first expression, 6:3::8:4, meant. The symbol √, which denotes a root, is without doubt derived from the letter r, which is the initial of root. And may not the symbol.., which usually denotes therefore, have been formed from the consideration that in a syllogism there are three essential parts or propositions, which are indicated in this symbol by the three dots placed in the above symmetrical form? V. The difference between two numbers which are composed of the same digits, the digits being placed in any order whatever, is exactly divisible by 9. For two numbers, being composed each of the same digits, will, when divided by 9, each leave the same remainder, hence their difference must be an exact multiple of 9, that is, it must be exactly divisible by 9. If from any number we subtract the sum of its digits, the remainder will be exactly divisible by 9. For we have already shown that any number divided by 9, will give the same remainder as will be found by dividing the sum of its digits by 9, therefore the difference between any number and the sum of its digits is exactly divisible by 9. From the above property, a very interesting arithmetical puzzle is deduced. Thus, you request a person to write down any number he may choose, then direct him to subtract from it the sum of all its digits; this done, request him to cancel or obliterate one of the digits of the result; after which, request him to give the sum of the remaining figures, and then you will be able to say just what figure was obliterated. For, had no figure been obliterated, the sum of the digits must have been divisible by 9, therefore the figure which it is necessary to add to the result which he gives you, to make it a multiple of 9, is the figure that was obliterated. If a nine or zero had been obliterated, the result would still be a multiple of 9, and you could then only say that it was either 0, or 9, that had been cancelled, but you could not tell which. The number 11 possesses some remarkable properties, which we will now proceed to point out. From the above method of separating the different powers of 10, we see that if 1 is added to the odd powers of 10, the results will be divisible by 11; also, if 1 be subtracted from the even powers of 10, the results will also be divisible by 11, since the numbers 11, 99, 1001, 9999, 100001, &c., are each divisible by 11. If we take any number, as 78534, it may, by the aid of what has just been shown, be decompounded as follows: 7853470000+8000+500+30 +4. But 70000 7 x 10000=7x (9999+1)=7× 9999+7 8x (1001-1)=8x1001-8 8000 8x1000 5x (99+1)= 5×99+5 3x11-3 +4 |