APPENDIX. I. NOTATION AND NUMERATION. It seems probable that the necessities of the human race would at a very early period suggest some method of counting or reckoning, as well as of registering the results of such processes: and the instruments employed, which in our language would be called Counters, might at any time convey to the mind a very distinct and clear idea of a number which did not consist of many individuals. Without entering into any historical account of the different Systems of Notation which have been used in different nations, or hazarding any conjectures as to the circumstances in which they may have had their origin, it is deemed sufficient for our present purpose to pass on immediately to the Notation now in use, which is fully explained in the first chapter of this work. 2. The characters 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are said to have been transmitted to us from the Arabians, who again are supposed to have received them from the Hindoos, though in forms considerably different from those in which they are now written: and as before hinted, the word Digit usually applied to them, denoting a Finger, seems to point out the means originally employed in estimating or computing numerical magnitudes, the number 10, which is called the Base or Radix of the system, and by which the local values of the digits are regulated, being that of the Fingers of both hands. The Notation appears to be as complete and convenient as can well be imagined, and in its present state may certainly be regarded as one of the greatest and most successful efforts of human ingenuity ever exhibited to the world. The reader who may be desirous of full information upon this subject is referred to Professor Leslie's interesting Work, entitled, The Philosophy of Arithmetic. 3. In reference to what was said in Article (14), it may be proper to observe that the method of proceeding differs from that adopted by the French and some other Foreign Arithmeticians, who adhere throughout to divisions of three figures, according to the principle of Article (11), and after the division of Millions, proceed directly to that of Billions, tens of Billions and hundreds of Billions: then to Trillions, tens of Trillions, and hundreds of Trillions, and so on: and this method certainly possesses some advantages in point of simplicity; but as numbers of these magnitudes are not of very frequent occurrence, it has not been thought necessary in the present performance to depart from the Notation and Nomenclature established in this country. In the following schemes it will easily be seen in what respects they differ. English Nomenclature. &c. 987654 321987 654321 where each division consists of six figures: and it may be extended towards the left hand as far as we please. French Nomenclature. each division consisting of three figures: and it is evident that as far as hundreds of millions are concerned, there is no difference whatever in the reading or enumerating of numbers in the two methods. II. ADDITION AND SUBTRACTION. 4. The very idea of number implies a capability of increase or decrease, the former of which is produced by the operation of Addition, and the latter by that of Subtraction and a set of Counters here represented by units will be of use in explaining the grounds upon which these operations are established. : Thus, suppose we wish to add five and seven together, then we have the following parcels of counters to represent them: 1, 1, 1, 1, 1, and 1, 1, 1, 1, 1, 1, 1; and if these be added together, or collected into one parcel, their sum will evidently be represented by 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, which may again be put in the form, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; and this indicates the result of the operation performed to be ten together with two, or twelve: that is, the sum of five and seven is twelve. Hence, in a system of counters there is in fact no operation to perform, except merely to collect or combine the counters into one group: and there would be no necessity for committing to memory the sum of two numbers as in our system, except so far as the name of that sum is concerned. The same might manifestly be done with more and larger numbers, and it furnishes the definition of the operation of Addition given in Article (21). 5. To subtract six from nine, implying that of nine individuals, six are to be taken away, we must evidently have at first nine counters, as 1, 1, 1, 1, 1, 1, 1, 1, 1, which may be formed into the two parcels, 1, 1, 1, 1, 1, 1, and 1, 1, 1; then if we withdraw six of these, or remove the first parcel, we have only three counters left, denoted by 1, 1, 1: and thus we see that if six be subtracted from nine, the remainder will be three. Here the withdrawal of the less number of counters, or the removal of the former parcel, is the only process employed, and of course it needs no effort of the mind to perform it: and our system has always a tacit reference to this circumstance, the operation of Subtraction being entirely founded upon it, and taking its definition from it, as in Article (26). 6. The student will perceive that the performance of these two operations is not facilitated by the modern notation, except as to the writing and reading of the results. On the contrary, they are rendered considerably more difficult, and require Rules and Directions to work by, which have already been laid down in Articles (22) and (27): they however depend upon a system of counters, owe their origin entirely to it, and may at any time be performed by means of it. Thus, retaining the use of the arithmetical signs for the operations of Addition and Subtraction, we have 3 = 1 + 1 + 1: 5 = 1 + 1 + 1 + 1 + 1: whence, 3 + 5 = (1 + 1 + 1) + (1 + 1 + 1 + 1 + 1) by omitting the brackets, which were introduced merely to keep the two parcels distinct from each other, and representing the aggregate or assemblage of units by its proper modern symbol: and it is here shewn, if need be, that it is quite immaterial in what order the numbers to be added are taken. Again, 7 = 1 +1 +1 +1 +1 +1 +1: 4 = 1 + 1 + 1 + 1 : whence, if from the former of these be withdrawn or removed what is equivalent to the latter, there will remain 1 + 1 + 1 or 3: and we shall have 7 – 4 = 3. 7. Although in the operations of Addition and Subtraction as treated of in the text, it has been found convenient to commence at the right hand and proceed towards the left, the use of the arithmetical signs will enable us to perform the same operations in any order we may choose: thus, to find the sum and difference of 1345 and 274, we have and the sum = 1000+500 + 110 + 9 = 1000 +500 + 100 + 10 + 9 From these processes, which have a close resemblance to the method of reckoning by counters, we cannot but see, that by a slight exercise of the mind and the memory, much real labour is saved by means of the rules in the text, not to mention the prolixity of operation as well as the number of figures that would be required for larger magnitudes than those which have been used to establish this conclusion. III. MULTIPLICATION AND DIVISION. 8. The operation intended by the word Multiplication, must necessarily be that which is defined and exemplified in Article (31) of the text, and it will here be required only to shew that the conclusions which it leads to, may be safely depended upon, as far as the order of the factors may influence the product. To multiply 4 by 3, we have to repeat 4 or 1 + 1 + 1 + 1, three times, and the product will therefore be (1 + 1 + 1 + 1) + (1 + 1 + 1 + 1) + (1 + 1 + 1 + 1) = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1) + (1 + 1 + 1), which is manifestly 1 + 1 + 1 or 3, four times repeated: that is, three times four is the same as four times three. |