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pursuits. When, in the course of these, you find yourself engaged in an inquiry that may seem to have but little relation to the things of sense, and are, on this account, disposed to ask-what is the use of it? just reflect whether the intellectual exercise has not been combined with intellectual gratification; and whether there be not an abstract beauty in the result arrived at, that awakens pleasurable feelings, practically realised, though hard to be described. We think the reflection will, in general, suggest an answer to the inquiry in some measure satisfactory.
These tranquil and purely intellectual enjoyments have prompted and encouraged the efforts of the wisest of men-have cheered and sustained them amidst penury and neglect; and, under the persecutions of power, wielded by ignorance, have supplied a consolation second only to that which Divine revelation affords; and they have finally been regarded as no mean reward for a life thus tried, and toils thus endured. They were considered as compensation sufficient for the labours even of NEWTON,-many of whose discoveries in science would, in all probability, never have been given to the world but for the urgent interference of private friends.
We are anxious that you should be influenced by considerations such as these; and that you should regard science as something more than a ministering agent to our animal comforts, or even to our social gratification and convenience. The practical benefits of science -and more especially of those departments of it connected with the subjects of the present volume-are in little danger of being overlooked or under-valued; they are spread profusely around us; and are felt and enjoyed by all. And there thus seemed to be all the more need for directing your attention to collateral advantages-less palpable and striking, and therefore less likely to be duly appreciated.
We shall now proceed to the business before us; we shall assume no knowledge at all on your part in reference to the topics to be discussed; and, in even so simple a subject as ARITHMETIC, we shall begin at the beginning.
Arithmetic.-Arithmetic is that branch of knowledge that teaches us how to perform calculations by means of numbers. The rules which direct the various operations constitute the art of arithmetic: the reasons and principles on which these are founded belong to the theory of arithmetic, and the theory and practice united, form the science of arithmetic. It is this that I am now going to explain. You are aware that the symbols, or marks, employed in this subject are called figures, and that they are as follow: 1, 2, 3, 4, 5, 6, 7, 8, 9, together with the mark 0, called nought, or cipher, or zero, and which stands for nothing. This 0 is also called a figure, so that there are ten figures in arithmetic: the number of units, or ones, which each stands for, is here written,—
Each of these figures is also called a number: but the word number has a wider meaning. Thus, 26, 43, 57, &c., are all numbers, each of which consists of two figures; the first number is twenty-six, the second forty-three, the third fifty-seven, and so on; so that, you see, the 2 in the first stands for two tens, the 4 in the second for four tens, and the 5 in the third for five tens. In like manner 368 is a number of three figures. The first figure, 3, stands for three hundred; the second, 6, for six tens, or sixty; and the third, 8, for eight ones, or units: the number itself standing for three hundred and sixty-eight.
You thus perceive that a figure which, when written singly, stands merely for so many units, changes its meaning, or value, according to the place it occupies in a number of several figures. If it occupy the last, or right-hand place, it still stands for units; but if it be in the next place, to the left, it stands for so many tens; if in the place next to that, for so many hundreds; and if it occupy the fourth place from the end, it stands for so many thousands: the number 7352, for instance, is seven thousand three hundred and fifty-two.
Figures thus have a local value, that is, a value depending upon the places they occupy in a number. The following is a number of twelve figures; and when the local values of these figures against them, it supplies what is usually called the
And this number is read thus,-two hundred and forty-six thousand eight hundred and seventy-three MILLION, one hundred and fifty-nine THOUSAND, four hundred and thirtyeight.
A person beginning to learn arithmetic will be enabled, by means of the above table, to read any number,—that is, to express its value in words. In a large number like that here given, the easiest way to proceed is this. Cut off the last three figures, then the next three, then the next, and so on; thus dividing the figures into sets of three as far as possible. A glance at the table shows that the leading figure of each set is hundreds of something; that of the first set, on the right, is hundreds of units, or simply hundreds ; that of the next set is hundreds of thousands; that of the next, hundreds of millions; and so on. And by thus finding out the local value of the leading figure in each period, as it is called, you may read the number with ease. For example, the number 68547329, when divided into periods, as here proposed, is 68,547,329: pointing to the 3, you say hundreds, and passing to the 5, hundreds of thousands; the incomplete period, 68, must therefore be 68 millions; and the entire number 68 million, 547 thousand, 329; or, expressing the value wholly in words, it is sixty-eight million, five hundred and fortyseven thousand, three hundred and twenty-nine. In a similar way we find the number 42638572613, or 42,638,572,613, to be 42 thousand 638 million, 572 thousand, 613. If you wished to put this wholly into words, all you would have to do would be to write forty-two for 42, six hundred and thirty-eight for 638, five hundred and seventy-two for 572, and six hundred and thirteen for 613. The leading figure of a complete period, you know, is always hundreds; and when you have found by the table what these hundreds are, or from practice can recollect what they are, you can have no difficulty in reading
the number. When any of the figures are noughts, a little extra care is, however, necessary. Thus, the number 460305007, which, divided into periods, is 460,305,007, is read four hundred and sixty million, three hundred and five thousand, and seven.
From what has now been said, you clearly see what is meant by the local value of a figure. If it occupy the place of units, its value is so many ones; if it be in the place of tens, its value is ten times as many ones, that is, it represents so many tens; if it be in the place of hundreds, its value is ten times as many tens; if in the place of thousands, it is ten times as many hundreds; and so on. So that, as you advance a figure, place after place, towards the left, you increase its value tenfold at every remove thus, 8 is simply eight ones; 80, where the 8 is now in the second place, and nothing, or nought, in the first place, is eighty, or ten times eight; 800, where the 8 is in the third place, and noughts in the first and second places, is eight hundred, or ten times eighty; and so on. This tenfold increase in the value of a figure, when it is removed one place from right to left, explains why our system of numeration is called the decimal system; the word decimal being derived from a Latin word, meaning ten. It was a beautiful contrivance thus to give a local, as well as an absolute value, to the symbols, or figures, used in the notation of arithmetic.* You see that by this happy idea we are enabled to express all numbers whatever by the help of only ten different marks, or symbols: whether we owe it to the Arabs, or to the Greeks or Romans, is a question on which there is still some doubt.
It ought, perhaps, to be mentioned, that although the numbers considered above do not extend beyond twelve figures, numbers with more figures than these may occur, and that there are words to express the additional periods. If the number have thirteen figures, the leading figure on the left would stand for so many billions; and if a complete additional period were joined to a number of twelve figures, making a number of fifteen figures, then the leading figure on the left would, of course, be hundreds of billions. But billions, trillions, quadrillions, &c., are names so seldom employed or wanted, that the numeration table need not be encumbered with them. It may be worth a passing notice, too, that no distinct ideas are conveyed by any of these terms; beyond a very moderate extent our notions of the value of numbers become confused. The number of ones in a million, even, is hard to conceive: it is a thousand thousand, and would take you more than twenty-three days to count, though you kept at it for twelve hours a-day, and counted one every second. Our ten figures, or digits, as they are often called (digitus being Latin for finger, and our ten fingers suggesting the word†),-our ten figures thus enable us accurately to express on paper, without the error of a single unit, numbers too great to be even conceived or imagined.
Before concluding these remarks on numeration, it may be as well to show the beginner how a number expressed in words may be translated into figures. This is not quite so easy as to translate figures into words; the plan is as follows:
Write down a row of noughts, or ciphers, and, as if these blanks were numbers, mark off the periods: then, commencing at the first cipher on the left, put under each the proper figure in the number proposed, taking care that it be in its proper place: if any vacancies appear under the corresponding ciphers, fill them up with noughts. Thus, let it be required to put into figures the number five hundred and six million, thirty-four thousand, and forty-eight. We know that the place of millions has six places to the
*The marks, or symbols, made use of in any science, constitute the notation of that science.
+ Names frequently throw light on the origin of things: it is interesting to notice that the name digit is plainly significant of the early rude method of counting on the fingers; and that the name calculation as plainly refers to the primitive practice of reckoning with pebbles (calculus, a pebble).
000 000 000
right of it; we therefore put a nought for the millions, and write six noughts after it, and, as we see, from hundreds being the leading word in the written expression, that the first period will be a complete period, we prefix two noughts more. The requisite row of noughts, divided as proposed, is as in the margin, and under these we now have to write, in their proper places, the figures 5, 6, 3, 4, 4, 8, and then to fill up the gaps with noughts; we thus find the number, when written in figures, to be 506,034,048. The learner will be able to do without such helps as these after a little practice; he should accustom himself to express in words the numbers he uses, when these are of moderate extent, and not content himself with merely looking at them.
506 034 048
I shall now proceed to the four fundamental operations of arithmetic. these are addition, subtraction, multiplication, and division. There are no calculations, however long and intricate, that are not composed of one or more of these four.
Simple Addition.—Addition teaches us how to add numbers together, and so to find the sum of all. It is called simple addition, when the numbers to be added either have no reference to particular things or objects, or when the things referred to are all of the same denomination: thus, if 24 pounds, 37 pounds, 82 pounds, &c., were all to be added together, the operation would be that of simple addition; but if 24 pounds 7 shillings, 37 pounds 2 shillings, 82 pounds 12 shillings, &c., were to be added, then, as pounds and shillings are different things, the operation would not be simple, but compound addition; one of the first set of things being called a simple quantity, and one of the other set a compound quantity. The rule for performing simple addition is as follows:
RULE. Arrange the numbers to be added one under another, so that the first column of figures on the right may be units, the next column tens, the next hundreds, and so on. This is nothing more than preserving each figure in its proper place. Add up the units' column: if it amount to a sum expressed by only one figure, put this figure down under the units' column. But if it be a number of more than one figure, the last figure only of that number—the units' figure—is to be put down, and the number expressed by what is left, after rubbing out the figure thus put down, is to be carried to the next, or tens' column, and added in with that column.
If the sum of the tens' column be a number of a single figure, it is to be put down under that column; but if it be a number of more than one figure, then, as before, only the last, or units' figure, of that number, is to be put down, and the number which is expressed, after the figure put down is rubbed out, is to be carried to, and added in with, the figures in the next column, and so on; observing, that when the last column is reached, the entire sum of that column is to be put down. Suppose, for example, the following numbers are to be added together,—namely, 246, 357, 26, 148, and 6; then, writing the numbers, one under another, as in the margin, so that the first column on the right may be a column of units, the next a column of ters, and the next a column of hundreds, we proceed, under the direction of the rule, as follows: 6 and 8 are 14, and 6 are 20, and 7 are 27, and 6 are 33; there are, therefore, in the first column, 33 units; that is to say, 3 tens, and 3 units: the 3 units we put, of course, under the column of units, but we carry the 3 tens to the next, or tens' column, and say,-3 and 4 are 7, and 2 are 9, and 5 are 14, and 4 are 18; that is 18 tens: the 8 we put down, but the number left, after rubbing this out, namely 1, we carry to the next column, as it is clear we ought to do; for this 1 is one more place to the left; it stands for one hundred, and therefore
belongs to the hundreds column: any figure next, on the left, to a figure that stands for tens, must, from the principles of numeration, stand for hundreds. Carrying, therefore, the 1 to the hundreds' column, we say, 1 and 1 are 2, and 3 are 5, and 2 are 7; that is, 7 hundreds: so that the sum of the proposed numbers is 783; that is, seven hundred and eighty-three. From this operation you see that the figures of the sum are all carefully put in their proper places, so that each has its own local value; the numbers carried from one column to the next, are so carried because they really belong to the place, one in advance, to the left. As a second example, let the numbers 8462, 873, 758, 4702, and 7003 be added together. Arranging these numbers one under another, as before, taking care not to disturb their local positions, we proceed thus: 3 and 2 are 5, and 8 are 13, and 3 are 16, and 2 are 18; 8 and carry 1: 1 and 5 are 6, and 7 are 13, and 6 are 19; 9 and carry 1: 1 and 7 are 8, and 7 are 15, and 8 are 23, and 4 are 27; 7 and carry 2: 2 and 7 are 9, and 4 are 13, and 8 are 21; therefore the sum is 21798: that is, twenty-one thousand seven hundred and ninetyeight. It is plain, from the foregoing illustrations, that the rule for addition is in strict accordance with the system of notation and numeration already explained, and that it must always lead to the correct result. There is no figure higher than 9: ten, of any denomination (hundreds, thousands, &c.), is one of the next higher denomination; so that in adding up any column of figures, all of the same denomination, for every ten in the sum, one must be carried to the next column; and, therefore, as many ones as tens.
873 758 4702
You have already seen that the marks used in the notation of arithmetic are figures : besides these, other marks are frequently employed to indicate operations with these figures, and to express relations among them: thus, instead of saying 2 and 5 are equal to 7, the form 2+5=7, is used to express the same thing: the mark+being the sign for addition, and the mark=the sign for equality: this must be borne in mind: +is called plus; so that 2+5=7, may be read 2 plus 5 equals 7, or 2 plus 5 are equal to 7. The following, therefore, are statements in symbols, instead of in words, which you will at once understand: 2+5+1=8; 3+4+2=9; 6+5+3+1=15. A few examples in addition are here given under this form. You will have to arrange the numbers in them in columns, as in the two examples worked above; and, if the results of your addition be correct, they will be found to agree with the numbers to the right of the sign of equality
Simple Subtraction.—Subtraction teaches us how to subtract the smaller of two numbers from the greater, or to find their difference, which is called the remainder. The operation is called simple subtraction when the numbers refer to things of the same denomination, as in simple addition. The rule for simple subtraction is as follows:
RULE. Put the smaller number under the greater, taking care, as in addition, that units shall be under units, tens under tens, and so on. Then, beginning at the units, subtract each figure in the lower row from the figure above it, if the lower figure be not the greater of the two, and put the remainder underneath (see the operation in the margin, where 34572 is subtracted from 68594, and the remainder found to be 34022). But if you come to a lower figure, which is greater than the figure above it, add 10