5. The Greek Notation.-At a very early period the Greeks had recourse to the twenty-four letters of their alphabet for the representation of numbers, and by means of these, aided by the three Hebrew characters 5, 5, A, they expressed the first three orders of numbers. Thus the numbers one, two, three, etc., to nine, were represented by • a, B, y, ô, ε, 5, 5, 1, 0 . . (1st order, or units). The numbers ten, twenty, thirty, etc., to ninety, by l, k, λ, μ, v, §, o, π, 5 (2nd order, or tens); •. and the numbers one hundred, two hundred, etc., to nine hundred, by P, o, T, v, P, X, Y, w, A (3rd order, or hundreds). Instead of multiplying distinct characters for higher numbers, they had recourse to their characters for the units, and by subscribing a small iota or dash, they denoted one thousand by a1, two thousand by B1, and so on. With these characters the Greeks could express every number under ten thousand. Thus 50 signified nine thousand nine hundred and ninety-nine, για In order to express higher numbers, they made use of the letter M, which, on being written below any character, increased its value ten thousand times. This contrivance enabled them to express all numbers as far as hundreds of millions; but instead of subscribing the letter M, it was afterwards found more convenient to write the letters Mu., or the contraction for μupia, ten thousand, after the character.whose value was to be increased ten thousand times; and then, when lower periods failed, they repeated the letters Mv. Thus Að Mv. Mu. Mv. signified thirty-four trillions. Archimedes, the most inventive of the Greek philosophers, divided numbers into periods of eight symbols each, which were called octades; and the famous Appollonius again divided them into periods of four symbols each, the first period on the left being units, the second myriads, the third double myriads, and so on. In this manner Appollonius was able to write any number which could be expressed by the present system of numeration. Having thus given a local value to his periods of four, it was remarkable that Appollonius did not perceive the advantage of making the period consist of less number of characters. Had he done the same thing with every single character, he would have arrived at the system now in common use, and this oversight is the more remarkable as the cipher was not unknown to the Greeks, but confined exclusively to their sexagesimal operations. 6. The Notation of the Romans.-The traces or strokes which originally represented numbers were replaced by those characters of the Roman alphabet which most nearly resembled them. The Roman notation was much ruder than the Greek, and for the expression of number they made use of the seven following capital letters, viz. : I for one; V for five; X for ten; L for fifty; C for a hundred; D for five hundred; and M for a thousand. By various repetitions and combinations of these they expressed all numbers. The four combined strokes which originally formed the character M for a thousand, assumed afterwards a rounded shape, fre quently expressed by the compound character CIO, consisting of the letter I inclosed on both sides by C, and by the same character reversed. This last form, by abbreviation on either side, gave two portions, one of which I was condensed into the letter D and expressed five hundred. The practice of using duodeviginti for octodecim, and so on, led the Romans to the application of deficient numbers; and instead of writing VIIII for nine, they counted one back from ten, and placing I before X, they wrote it thus, İX. In a similar manner XIX represented nineteen, XL forty, XC ninety, and CM nine hundred. They also repeated the symbols of a thousand to denote higher numbers; thus CCIƆƆ represented ten thousand, and CCCƆ an hundred thousand. Separating each of these, gives 100 for five thousand, and IƆƆƆ for fifty thousand. Also a horizontal line drawn over any letter augmented its value one thousand times; thus LX signified sixty thousand. With this explanation the following examples will be readily understood: I one. V five. VI six. XIII thirteen. XIV fourteen. XV fifteen. XVI sixteen. XVII seventeen. VII seven. XVIII eighteen. X ten. XXX thirty. XI eleven. Thus as often as any symbol is repeated, its value is repeated as often; chapters and sections of LXX seventy. XC ninety. C one hundred. X ten thousand. 7. Although the Greek arithmetic, as successively moulded by the ingenuity of Archimedes, of Appollonius, and of others, had attained to a high degree of perfection, and was capable of performing operations of very considerable difficulty and magnitude, still the great and radical defect of the system consisted in the entire absence of a general mark corresponding to our cipher, which without having any value in itself, should yet serve to keep the rank or power of the other characters, by occupying the vacant places in the scale of numeration. From the preceding remarks on the notation of the Greeks and Romans, the student will be able to form some idea of the great superiority of the present system, which has led to some of the most striking and remarkable scientific discoveries. 8. In the common system of numeration all numbers, however large or small, can be expressed by the ten following characters or figures, viz. : 1 2 3 4 5 6 7 8 90 one, two, three, four, five, six, seven, eight, nine, nothing. The first nine of these are called significant figures or digits, and sometimes represent units, sometimes tens, hundreds, or higher classes. When placed singly, they denote the simple numbers subjoined to the characters; when several are placed together, the first figure on the right is taken for its simple value, the next signifies so many tens, the third so many hundreds, and the others so many higher classes corresponding to the order in which they are placed. Thus 4532 signifies four thousand, five hundred, thirty, and two units ; and in the number two hundred and twenty-two, which is written thus 222, the figure 2 is repeated thrice, but each has a different value; the first on the right hand is two units, the second two tens or twenty, and the third iwo hundreds. 9. When any of the denomivations, units, tens, hundreds, etc., is wanting, it becomes necessary to supply its place with the last sign or character, viz., 0, which is termed cipher or nothing, the word cipher in the Arabic language signifying vacuity. This character which indicates the absence of all number, is a most important one, inasmuch as its introduction serves to preserve the proper positions of the sig, nificant figures. Thus the number forty thousand three hundred and twenty would be expressed in figures by 40320, because the denominations, units, and thousands are wanting, and the absence of each is indicated by the cipher which occupies its place. From these illustrations we may perceive that the superiority of our present system of numeration arises from a few simple signs being made to change their value as they change the position in which they are placed, and that the significant figures have a local as well as a simple value. It is thus that, in consequence of the established relative value of units and tens, the same figure which, beginning on the right, expresses units, becomes ten times greater at each remove to the left, and by simply changing their places, the different characters become susceptible of representing successively all the different collections of units which can possibly enter into the expression of a number. Thus we get10 11 12 13 14 15 19 sixteen, seventeen, eighteen, nineteen; where the first figure on the left signifies ten, ånd the second figure its simple value, or so many units. Hence 10 means ten and nothing; 11 ten and one, and so on. Again, 20 means two tens and nothmy, or twenty; 21 two tens and one, or twenty-one ; 30, thirty; 90, ninety; 100, ten tens or one hundred ; and 1000, one thousand. The names and values of numbers will be readily acquired from the following examples. The figures which compose a large number are separated into periods and half-periods, for the more readily ascertaining the precise position which each figure occupies. The period consists of six figures, and the first, beginning on the right, is called the period of units, the second the period of millions, the third the period of billions, a contraction for millions of millions or bi-millions, and so on. Thus the number Trillions. Billions. Millions. Units. 490,386; 407,135; 017,693; 125,076 is read thus :-Four hundred and ninety thousand three hundred and eighty-six trillions; four hundred and seven thousand one hundred and thirty-five billions ; seventeen thousand six hundred and ninety-three millions; one hundred and twenty-five thousand and seventy-six. 10. There does not appear to be any number naturally adapted for constituting a class of the lowest or any higher rank to the exclusion of others; though it is very probable that our present system of numeration had its origin in the practice of reckoning with the ten fingers. The number ten is called the radix or scale of the common system, because in it we ascend by collections of ten in each class to the next higher class, and though almost all nations have adopted this number as the base of their system of numeration, still it is perfectly arbitrary, and convenient as it may be for general use, there may be other scales, such as the duodenary, whose base is 12, which possess superior advantages. But whatever be the scale of notation made use of, the same principle will enable us to write all numbers in that scale. Thus in the quinary scale whose radix is 5, we need only the five characters, 0, 1, 2, 3, 4, and each figure placed on the left of another will have a value five times greater than if it occupied the place of this last. Hence in this scale, 10 means five, 11 six, 12 seven, and so on. In this as well as in every other scale, except the denary or decirnal one, we find a difficulty in enunciating a number, because there is no longer an accordance with the decimal language which pervades the construction not only of our own, but of all civilized languages. 11. Numbers may be viewed in two ways, either by considering them without particularizing the unit to which they refer, or by designating what they are intended to enumerate. Thus two, three, five, seven are abstract numbers, while three men, five days, seven books are concrete numbers. It is evident that the formation of numbers, by the successive re-union of units, does not depend upon the nature of these units, since 5 days and 7 days together make 12 days, 5 acres and 7 acres together make 12 acres, and 5 and 7 together make 12. 12. Since numbers can only be changed by increasing or diminishing them, it follows that the whole art of arithmetic is comprehended in two operations, which are termed Addition and Subtraction. But as it is frequently required to add several equal numbers together, as well as to subtract several equal sums from a greater, till it be exhausted, other methods have been devised for facilitating the operation in these cases, and named respectively Multiplication and Division. These four rules are the foundation of all arithmetical operations whatever. ADDITION. 13. Addition is the collecting together of two or more numbers, and the amount of all of them is termed their sum. The sign + (plus) is employed to indicate addition, and 7 + 2 signifies that 2 is to be added to 7. Also the sign (equal) signifies that the numbers between which it is placed are equal : thus 8 + 1 = 9. EXAMPLES 1. Let it be required to find the sum of the two numbers 1724 and 4638. Take them to pieces, separating them into thousands, hundreds, tens, and units. Thus 1724 = 1 thousand, 7 hundreds, 2 tens, and 4; 4638 = 4 thousands, 6 hundreds, 3 tens, and 8. To each of the four parts into which the first number is separated add the part of the second which is under it, beginning at the units. Thus 8 units and 4 units are 12 units; that is, 1 ten and 2 units; again, 3 tens and 2 tens are 5 tens; 6 hundreds and 7 hundreds are 13 hundreds, or I thousand and 3 hundreds. Lastly, 4 thousands and 1 thousand are 5 thousands; hence the sun is either 5 thousands, 13 hundreds, 5 tens and 12 units; or 6 thousands, 3 hundreds, 6 tens and 2 units = 6362. 2. Let it be required to find the sum of 26389, 38127, 2815, 6817, 490, 25 and 3745. Write the numbers, as at the side, so that the figures of the same class shall be in the same vertical columns. Then taking the 26389 sum of each class, we find there are 38 units, 27 tens, 31 38127 hundreds, 25 thousands, and 5 tens of thousands. Now 2815 38 units are 3 tens and 8 units; then writing 8 below the 6817 units' column, carry the 3 tens to the 27 tens, which together 490 make 30 tens, or 3 hundreds and 0 tens. Write 0 below the 25 column of tens, and reserve the 3 hundreds to be added to the 3745 31 hundreds. This gives 34 hundreds or 3 thousands and 78408 4 hundreds, and writing 4 below the column of hundreds, carry the 3 thousands to the 25 thousands, and we get 28 thousands, or 2 tens of thousands and 8 thousands. Writing 8 below the column of thousands, carry the 2 tens of thousands to the 5 tens of thousands, and finally write 7 below the column of tens of thousands, making the entire sum = 78408. 14. From these principles the following rule may be drawn : Rule. Write the numbers to be added together in vertical columns, 80 that the units of all the numbers may be in one column, the tens in the second, the hundreds in the third, and so on. Draw a line under the last number, and, beginning with the column of units, add successively the numbers contained in each column: if the sum does not exceed nine, write it down under the line; but if it contain tens, reserve them to be added to the next column, writing down only the units of each column; and under the last column put the entire sum whatever it may be. If the sum of any column be an exact number of tens, write 0 for the units and carry the tens to the next column. 15. The results of the partial additions being furnished by the memory, it is desirable to have some plan of testing the accuracy of the final sum, and this may be done in various ways; but we shall only mention the two following methods of the proof of addition : 1. Having found the sum in the usual way, begin at the top and |