equality was introduced by Record, the above-named English mathematician, and for this reason, as he says, that “noe 2 thynges can be moar equalle," namely, than two parallel lines. It is a curious circumstance that this same symbol was first used to denote subtraction. It was also employed in this sense by Albert Girarde, who lived a little later than Record. Girarde dispensed with the vinculum employed by Steifel, as in 3+4, and substituted the parenthesis (3+4), now so generally adopted. The first use of the St. Andrew's cross, X, to signify multiplication is attributed to William Oughtred, an Englishman, who in 1631 published a work entitled Clavis Mathematicæ, or Key of Mathematics. It was intended to notice several other works, ancient and modern, but the length to which this sketch has already extended forbids it We must not, however, omit to mention two American works, which have done much for the cause of practical Arithmetic in this country. These are the large work of Nicholas Pike, first published about 1787, and the little unpretending - First Lessons ” in Arithmetic, by War ren Colburn. From the former of these many later authors have borrowed much that is useful, and the latter has exerted an influence on the method of studying Arithmetic greater, perhaps, than any other modern production. No better elementary work than that of Colburn has ever, it is believed, appeared in any language. We had thought of alluding to the ancient philosophic Arithmetic, and the elevated ideas which many of the early philosophers had of the science and properties of numbers. But a word must here suffice. Arithmetic, according to the followers of Plato, was not to be studied “ with gross and vulgar views, but in such a manner as might enable men to attain to the contemplation of numbers ; not for the purpose of dealing with merchants and tavern-keepers, but for the improvement of the mind, considering it as the path which leads to the knowledge of truth and reality.” These transcendentalists considered perfect numbers, compared with those which are deficient or superabundant, as the images of the virtues, which, they allege, are equally remote from excess and defect, constituting a mean between them; as in the case of true courage, which, they say, lies midway between audacity and cowardice, and of liberality, which is a mean between profusion and avarice. In other respects, also, they regard this analogy as remarkable: perfect numbers, like the virtues, are “ few in number and generated in a constant order; while superabundant and deficient numbers are, like vices, infinite in number, disposable in no regular series, and generated according to no certain and invariable law." We conclude this brief sketch in the earnest hope that the noble science of numbers may ere long find some devoted friend who shall collect, arrange, and bring within the reach of ordinary students, much more fully than we have done, the scattered details of its longneglected history. ARITHMETICAL SIGNS. = Sign of equality; as 12 inches = 1 foot signifies that 12 inches are equal to one foot. + Sign of addition; as 8+6= 14 signifies that 8 added to 6 is equal to 14. - Sign of subtraction; as 8—6= 2, that is, 8 less 6 is equal to 2. x Sign of multiplication; as 7 x6 = 42, that is, 7 multiplied by 6 is equal to 42. Sign of division; as 42 = 6=7, that is, 42 divided by 6 is equal to 7. 72 Numbers placed in this manner imply that the upper num ber is to be divided by the lower one. :::: Signs of proportion ; thus, 2:4::6: 12, that is, 2 has the same ratio to 4 that 6 has to 12; and such numbers are called proportionals. 12—3+4= 13. Numbers placed in this manner show that 3 is to be taken from 12, and 4 added to the remainder. The line at the top is called a vinculum, and connects all the numbers over which it is drawn. 92 implies that 9 is to be raised to the second power; that is, multiplied by itself. 83 implies that 8 is to be multiplied into its square, or to be raised to the third power. ♡ This sign prefixed to any number shows that the square root is to be extracted. Ñ This sign prefixed to a number shows that the cube root is to be extracted. Sometimes roots are designated by fractional indices, thus; 9* denotes the square root of 9; 27$ denotes the cube root of 27. ( [] Parentheses and brackets are often used instead of a vinculum. Thus, (7 — 3) x 5= 60 3. 1 An edition of this work, without answers, is published for the accommodation of those teachers who prefer that the pupil should not have access to them. A Key, containing solutions and explanations, is also published for the convenience of teachers. ARITHMETIC. SECTION I. ARITHMETIC is the science of numbers, and the art of computing by them. The operations of Arithmetic are performed principally by Addition, Subtraction, Multiplication, and Division. NUMERATION. NUMERATION teaches to express the value of numbers, either by words or characters. Nunibers in Arithmetic are expressed by the ten following characters, which are called numeral figures ; viz. 1 (one), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (seven), 8 (eight), 9 (nine), o (cipher, or nothing). The first nine of these figures are called significant, as distinguished from the cipher, which is of itself insignificant. Besides this value of the numerical figures, they have another value, dependent on the place which they occupy, when connected together. This is illustrated by the following table and its explanation. Here any figure occupying the first place, reckoning from right to left, denotes only its simple value or number of units. But the figure standing in the second place denotes ten times its simple value ; that occupying the third place, a hundred times its simple value, and so on to any required number of places ; the value of any figure being always increased tenfold by its removal one place to the left. Thus, in the number 1834, the 4 in the first place denotes only four units, or simply 4; the 3 in the second place signi. fies three tens, or thirty; the 8 in the third place signifies eighty tens, or eight hundred ; and the l in the fourth place, one thou. sand; so that the whole number is read thus, – one thousand eight hundred thirty-four. Although the cipher has no value of itself, when standing alone, yet, being joined to the right hand of significant figures, it increases their value in a tenfold proportion; thus, 5 signifies simply five, while 50 denotes five tens, or fifty ; 500, five hun. dred, and so on. Note. - The idea of number is the latest and most difficult to form. Before the mind can arrive at such an abstract conception, it must be familiar with that process of classification, by which we successively remount from individuals to species, from species to genera, from genera to orders. The savage is lost in his attempts at numeration, and significantly expresses his inability to proceed by holding up his expanded fingers or pointing to the hair of his head. See Lacroix NUMERATION TABLE. The following is the French method of enumeration, and is in general use in the United States and on the continent of Europe. In order to enumerate any number of figures by this method, they should be separated by commas into divisions of three figures each, as in the annexed table. Each division will be known by a different name. The first three figures, reckoning from right to left, will be so many units, tens, and hundreds, and the next three so many thousands, and the next three so many millions, &c. Vigintillions. The value of the numbers in the an. nexed table, expressed in words, is One hundred twenty-three vigintillions, four hundred fifty-six novemdecillions, seven hundred eighty-nine octodecil. lions, one hundred twenty-three septendecillions, four hundred fifty-six sexdecillions, seven hundred eighty-nine quindecillions, one hundred twenty-three quatuordecillions, four hundred fifty-six tridecillions, seven hundred eighty-nine duodecillions, one hundred twenty-three undecillions, four hundred fifty-six de. cillions, seven hundred eighty-nine nonillions, one hundred twenty-three oc. tillions, four hundred fifty-six septillions, seven hundred eighty-nine sextillions, one hundred twenty-three quintillions, four hundred fifty-six quadrillions, seven hundred eighty-nine trillions, one hundred twenty-three billions, four hundred fifty-six millions, seven hundred eightynine thousands, one hundred twentythree units. |