CHAPTER 1. THE NUMERAL CHARACTERS, OR, ELEMENTARY PRINCIPLES OF INCREASE AND DECREASE.* The idea of numbers is one of the first that enters into the human mind. The infant observes his two hands, or the two eyes of his mother. The rudest savage counts his arrows or his game. Names for numbers, therefore, are among the first words invented. A few names, of course, answer every purpose for the child or the savage. But, as the child becomes a man, or the savage becomes civilized, new wants call for new numbers, and these of course call for new names, until it becomes impossible to supply a sufficient variety. For no genius could invent, no memory retain, such a multitude of terms, were a distinct word required for each number. Hence mankind has everywhere been compelled to classify numbers. Thus, if sixty or a hundred shells were spread out on the seashore, or placed in a row, to explain to a group of savages the number of fish contained in a canoe, the collection or row would not give a clear idea of the actual quantity. But, if the shells were arranged in small heaps, each of which should contain an equal number, there could be no such difficulty. Now, such an arrangement has been actually introduced into every community. Nature herself has provided us with a scale, or measure, which is so obvious and simple as to have forced itself into universal use. This is no other than the ten fingers. By the aid of this scale any number whatever can be expressed by the aid of a very small number of terms. Nor is this the only advantage of this scale. It requires only nine characters, with an additional one to express zero, or nothing, to represent this wonderful, this infinite variety of numbers. * If the class of beginners is young, it would be profitable for the teacher to read this chapter to them by sections, with illustrations on the black-board and other explanations. Ten, then, being the universal scale or measure used in calculation, our system is properly called Decimal Arithmetic, the word decimal meaning numbered by tens. Various kinds of characters have been used at different times, and by different nations, for expressing numbers. But the Roman and the Arabic numerals are the only ones which it is important for the student of Arithmetic to understand. The Roman numerals are chiefly used for dates, chapters, and sections, of books, and the hours on time-pieces. The characters are derived from the alphabet. Their origin is sufficiently evident; and, as a knowledge of this origin will assist the student in recalling them to mind if they should be forgotten, an explanation of it will not be out of place here. The ten fingers present so obvious and convenient a method of numbering, that every people hitherto known, except the Chinese, and an obscure tribe mentioned by Aristotle, has employed them for that purpose. The rude tribes of Africa and America, however, use the fingers of one hand only as their scale; that is, they count onward from one to five, as we do from one to ten, and then commence anew. It may justly be affirmed, then, that nature, in forming the human hand, supplied us, at the same time, with the first elements of calculation. But the Romans not only used the digits, or fingers, as the foundation of their method of computing; they also derived several of their characters from them. Thus, a finger, repre ented by I, stood for one ; two, three, and four fingers, represented by II, III, IIII, stood for two, three, and four. By holding up the hand with all the five fingers extended, a tolerably correct representation of the letter V will appear, formed by the thumb and index finger. V was accordingly chosen as the character for five. In like manner, VI (six) is one hand and one finger of the other ; VII (seven) a hand and two fingers, &c., while X (ten) represents both hands, considered as two V's, joined by their apices ; or it may be formed by holding up both hands, one thumb resting on the other in the form of a cross. C and M, the initial letters of centum and mille, the Latin words for a hundred and a thousand, represented these numbers. C was originally written thus, [. Its half L, stood for 50. In like manner the half of M, N, rounded into D, stood for 500. Such was evidently the origin of the first Roman numerals; but, as the eye does not readily recognize more than three characters at a glance, a plan has been adopted to obviate that difficulty, and that is, by causing a smaller number placed before a larger to be subtracted in place of being added. Thus, in place of IIII (four times one), we have IV (five less one); for VIIII (five and four), we have IX (ten less one); for XXXX (four times ten), we have XL (fifty less ten); and for LXXXX (fifty and forty), we have XC (a hundred less ten.) Besides the characters already enumerated, 15 is sometimes used for D, and CIÒ for M; and these, in fact, may possibly be the original characters that represented five hundred and a thousand. For, when brought closely together, they greatly resemble the D and the M. But in other respects they are out of rule. For, when is annexed to Iā, it increases the value of the latter tenfold. In like manner, when C is prefixed and O annexed to CID the last is increased tenfold. Lastly, the value of a character is increased a thousand fold by drawing a horizontal line over it. TABLE OF ROMAN NUMERALS. I. one. CCC. three hundred. CD. four hundred. DCC., or IDCC., seven hundred. DCCC., or IOCCC., eight hundred. VII. seven. CM. nine hundred. VIII. eight. M., or CI5., a thousand. MMM., or III, three thousand. MMMM., or IV., four thousand. Ibb., or V., five thousand. L. fifty IDOM., or VI., six thousand. LX. sixty. IDOMM., or VII., seven thousand. IOOMMM., or VIII., eight thousand. IOOMMMM., or IX., nine thousand. &c. &c. Every one must see what a tedious affair a large calculation would be according to this cumbrous system of notation; nor is it easy to say what our commercial standing, to say nothing of science, would have been to-day had it never been superseded. Exercises for the Black-board and Slate. 1. Write the following numbers in words, explain each letter separately, and, lastly, read the whole series in connection : MDCCCLIV; MMMCCLX; IIIXL; XCIV; CMXCIX; XIDCLII; XX; XIX; MMDXXXII; IIIDXXV; XCVII; IOCXXXVIII; XXIICDLVI; LVIII; LXXXVII; XLVII; XVI; XIX; XXIV; XXXIX; IIV; XX; XCIX; XXVIII; XXIX; LVI; CCCLXIV; XXV; XVI; IV. 2. Write the following numbers in Roman numerals, read them, and explain each separately. Eighteen hundred and fifty-four; eighteen hundred and nineteen; twelve thousand two hundred and sixty; three thousand and forty; ninety-nine ; fifty-four; ninety-four; fortysix; nine hundred and ninety-nine ; ten thousand six hundred and fifty-two; ten thousand and ten; forty-nine ; eighteen thousand seven hundred and thirty-six. Questions to be put by the Teacher. What does the I represent? Ans. A finger. The V? The X? The C? The L? The M? The D ? &c., till all the characters in the table are explained, and their origin pointed out. The Arabic Numerals, as they are called, though they are now generally allowed to be of Indian origin, were introduced into Europe by the Arabs nearly a thousand years ago. They are now used by all civilized nations. The Arabian method unites the important advantages of conciseness, simplicity, and precision. Indeed, it is impossible to conceive anything better adapted to the purposes of calculation. A more convenient scale than that of ten might have been adopted, so as to have allowed of more equal subdivisions without fractional parts (for instance, the scale of eight, of sixteen, or of twelve); but the principles of the notation are incapable of improvement. The number of characters in the Arabic notation is ten. Nine of these represent numbers, and one stands for nothing, by itself, though indispensable to the system. The Arabic characters have also probably originated from the fingers. But they differ from the Roman numerals in this, that some of them consist of vertical, others of horizontal lines, and others again of both. The following are the characters, with their names. Underneath each is placed their supposed original form: one two seven 1 TABLE OF ARABIC NUMERALS. eight nine nought, or cipher. Thus, one is represented by a vertical line, as in the Roman system; two by two horizontal lines; three by three of the same; four, by a square, that is, two horizontal and two vertical lines; five, by three horizontal and two vertical; six, three orizontal and three vertical; eight (two fours), two squares; seven, two squares, less one vertical; nine, evidently borrowed from the Greek character for nine (9, theta.) The seven is also supposed to be borrowed from the Greek character for that number (5, zeta), to which it certainly bears considerable resemblance. Lastly, the nought, or cipher, which does not consist of lines to be counted like the others, but, on the contrary, is entirely round, to show that of itself it has no value. The first nine characters have been rounded to their present form, doubtless, by rapidity in writing. Formerly, the ten Arabic characters were all called ciphers, from the Arabic word sipher, to enumerate. Hence, arithmetic is often called ciphering. The first nine are now called digits, a name derived from the Latin word digitus, which signifies a finger. They are also called significant figures, because each of them has a peculiar value of its own, and to distinguish them from the cipher, which has no value of itself, though it is an exceedingly important figure, as it often modifies the value of all the other figures, as will presently appear, By means of these ten characters, any number can be expressed, however small or great it may be. This is effected by affixing two kinds of value to each of the significant figures, namely, their primary, or simple, or absolute value, and their secondary or local value. Their simple value is always the It is expressed by their names as given above. The local value differs according to the place, or rank, which the same. |