L known, except the Chinese, and an obscure tribe mentioned by Aristotle, have had recourse to them. The Romans not only used the digits, or fingers, as the foundation of their method of computing, but also derived from them several of their characters. Thus, a finger, represented by I, stood for one; two, three, and four fingers, represented by II, III, and 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. In like manner, VI is one hand and a finger of the other; VII, a hand and two fingers, &c.; while X represents both hands, considered as two Vs, joined by their apices; or it may be formed, by holding up both hands, one thumb resting across the other. Č and M, being the initial letters of centum and mille, the Latin words for 100 and 1000, stood for those numbers. and D, representing half of C and M,* stand for their halves, 50 and 500. This was undoubtedly the arrangement of the first Roman numerals; but, as the eye does not readily distinguish more than three similar characters, at a glance, a plan was adopted, to prevent the recurrence of more than three, by making a smaller number, when placed before a larger, to be subtracted in place of added. Thus, in place of IIII, we have IV5 less 1; for VIIII 5+49, we have IX=10 - 1 = 9; for XXXX 4 tens, we have XL=50-10; for LXXXX 50+40=90, we have XC : 100-1090. When this form of numeration has been explained to a class, they ought to be questioned on the blackboard, till the subject has become perfectly familiar. A somewhat complicated question is subjoined, as a specimen. T. What number is MDCCCXLIV ? C. M is 1000; D, 500; three Cs, 300; together 1800; X, 10, to be taken from L, 50, leaves 40; I, 1, from V, 5, leaves 4 altogether, 1844. T. What does the I represent? C. A finger. *The C was originally written, C, half of which is L; half the M, A, only wants a little rounding, to transform it into D. T. The V? C. A hand. T. The X? C. Two hands. T. The C? the L? the M? the D? The third system of characters representing numbers, is that which is commonly called the Arabic, though now generally allowed to be of Indian origin. It was introduced into Europe about A. D. 1130. This is so very much superior to all others, that it is hardly credible that our long and complicated arithmetical operations could have been performed before it came into use. The characters in this system indicating numbers, (commonly called the significant figures,) are only nine. They have most probably received their origin, also, from the fingers, but in a different manner from the Roman numerals; some of the marks consisting of vertical, others of horizontal lines. The following are supposed to be the original forms of the characters: Thus, 1 is represented by a vertical line, as in the Roman system; 2, by two horizontal ones; 3, by three do. ; 4, by a square, or two vertical and two horizontal lines; 5, by three horizontal and two vertical; 6, three horizontal and three vertical; 8, (two fours,) two squares; 7, two squares, less one vertical; lastly, 9, evidently borrowed from the Greek 9, (9, theta.) The 7 also is, by some, supposed to be borrowed from the Greek , (zeta,) to which it bears a considerable resemblance. All the characters have been rounded to their present form, by rapidity in writing. These nine characters have each two values, viz., their simple value, as one, two, three, &c., and their local value, which depends on their distance from the place of units, which is always the first on the right hand, unless otherwise indicated by a mark, which shall be explained presently. Thus, in the following number, 6666, we Brown. have six, four times repeated; but every time the character represents a different value, the first on the right hand representing the units, (or ones,) and, therefore, simply six; the second, 6 ty, or tens; the third, 6 tens of tens, or hundreds; the fourth, 6 tens of tens of tens, or thousands; and, if there were more, they would still go on, increasing tenfold, to infinity. Thus we perceive, that the fundamental law of the Arabic system is, that a removal of a figure one place towards the left increases its value, tenfold; and, on the contrary, its removal towards the right decreases it, tenfold. In addition to the nine characters mentioned above, there is one which does not consist of lines, like the significant figures, but, on the contrary, is entirely round, to express that it has in itself no value, its sole use being to occupy the place of some denomination, which may be wanting, and which, therefore, instead of its customary name of cipher, may be appropriately termed figure of place. Thus, to represent six hundred and five, (605,) it is necessary to have a character that has no value, in itself, to stand in the place of tens; otherwise, the 6 would be 6 tens or ty, in place of 6 hundred. Whenever the figures representing any number consist of more than four, they should be divided, by commas, into series of threes, the unit being counted as the first, whether they extend to the right or left of it. Every figure, besides its simple name, (one, two, three, &c.,) has two other names, which may be likened to the Christian and family names of children, as exemplified in the following table : Smith. Tens. John Units. Bill Hund. Tom Tens. John Units. Bill • Hund. Tom & Tens. John Units. Carrying out our simile, we may say, that, in every family, excepting in the one on the extreme left, there must always be three chairs for the boys, none of which must ever be empty; for, if any boy be absent, a block (cipher, or figure of place, 0) must occupy his chair, till he returns. For the sake of brevity, we never name any of the Johns, nor the family name of Jones. For instance, instead of saying John Brown, John Smith, &c., we merely say Brown, Smith, &c., and Bill Jones and Tom Jones are merely called Bill and Tom. Applying this to the real local names of the figures, we never make use of the word units, at all. Whenever either or both local names are wanting, units is always understood. For instance, when we say, two, we mean two units of units, and when we say six thousand, we mean, six units of thousands. Let, now, a series of figures be written on the blackboard; let one of the class divide them into series of threes, and exercise the class in naming them, irregularly, till all become quite familiar with the subject. Again, write several figures of the same kind, thus : 6 5 4 3 2 1 4 4 4 4 4 4, and ask, how many times is the second 4 greater than the first? how many times is the third greater than the first ? the fourth, than the first? the fourth, than the second? the fourth, than the third? &c. Again, how many times is the first contained in the second? the first in the third ? the second in the sixth ? &c. T. What is the name of the first series of threes? T. The third? C. Millions. T. The second? the fourth? &c. T. Repeat the three names of the first series. C. Units, tens, hundreds. T. Repeat the three names of the third series. Are they the same in every series? T. What name is never used? C. Units. Again, write 4060 T. How much is the 4? the 6? What is the use of the figures of place? C. To show the place of the 4 and the 6. T. If I remove the 6, will the value of the 4 be changed? Will the value of the 6 be changed, if I remove the 4? Why not? C. Because, as we name from the right, the 6 will still stand in the place of tens of units, whatever change we make in the figures on its left. T. If I place a 6 on the right of the sum, thus, 40606, are the other 6 and 4 changed, and how? T. Fifty-two millions, six thousand, and twenty. How many figures are necessary to represent this sum? How many of them are significant figures? Questions similar to the above, having been repeated, till the notation of integers has become quite familiar to the class, the teacher may turn to the article Arithmetic, in Chap. V. of Part I., of this book, where directions are given, how to explain and comment on notation of decimals. The class will then take their books and slates, and proceed to express, in figures, numbers given in words, using all the examples given in the books of arithmetic, both in integers and decimals. Should the class not be familiar with the practice of rendering words into figures, after they have exhausted all the questions in their book of arithmetic, the teacher should make new ones, or procure them from some other book. The pupils can never be expert in arithmetic, till this subject is perfectly understood. Addition.—This rule is perhaps more used in business than any other in arithmetic. The student, before he leaves it, should be able to add long sums, rapidly and correctly, such as are found in merchants' books, &c. If he has practised mental arithmetic, long sums may be given him at once; if not, he may commence with shorter ones, and gradually lengthen them. As soon as he acquires a little proficiency, he should begin to add two or |