PART I. ARITHMETIC. § 1. 1. MATHEMATICS is the science of Quantity. 2. QUANTITY is a term used to denote whatever admits of being increased, diminished, or measured. 3. NUMBERS are certain expressions for determinate quantity. § 2. 1. ARITHMETIC is a branch of mathematics which regards the properties and combinations of numbers. 2. Its principal operations are six; NOTATION, NumeraTION, ADDITION, MULTIPLICATION, SUBTRACTION, and Divi SION. 3. Four of these are essential and distinct; Notation, Numeration, Addition and Subtraction: the remaining two, Multiplication and Division, being embraced in two of the others, Multiplication in Addition, Division in Subtraction, are not essentially distinct operations, but are so treated for convenience. I. NOTATION. § 3. 1. NOTATION is the writing of numbers. 2. It has two methods; one by characters or figures; the other by letters; known as the Arabic and Roman. 3. That by characters or figures, is the Arabic; that by letters, the Roman.* *These methods are thus called because of their having been derived respectively from Arabia and Rome. The Arabians were early devoted to scientific pursuits, which involved mathematics, and so were led to devise convenient and concise methods of expression. How early they invented and adopted the present characters is not accurately ascertained; but they were introduced into Europe in the ninth century. The Romans devoted themselves rather to literature, and so were under no necessity to seek out a better mode of representing quantity than that which they have given us. § 4. 1. The Arabic method of Notation, for its conciseness and convenience, is that in most common use. 2. It employs ten figures, by various combinations of which, all determinate quantities may be expressed. 3. These figures are distinguished as significant and insignificant. The former, sometimes called digits,† are 1, 2, 3, 4, 5, 6, 7, 8, 9: the latter, usually called cipher, is 0. Significant means with value; insignificant without value. § 5. 1. Each figure has its own specific or simple, and its local value. 2. The simple value of a figure is that which it has when it stands by itself. 3. The local value is a value that varies, according to the place in which the figure stands when combined. In a combination of figures, reckoning from right to left, the figure in the first place represents its simple value: those in the succeeding places have a local value. § 6. 1. Without combination, we cannot express more than nine. 2. To denote ten, a combination is made of the first significant figure (1) and cipher (0); the 1 being placed at the left of 0; thus, 10. 3. The increase is continued up to nineteen, by placing the significant figures at the right of the 1, in the room of the 0: thus, 11, 12, 13, 14, 15, 16, 17, 18, 19. 4. It is continued one farther, by increasing the figure at the left, and placing 0 at the right of it; thus, 20. 5. To continue the increase further, the significant figures are written in the same manner as before, (§ 6. 3); thus, 21, 22, 23, &c.: and so on up to 99. 6. After the nine digits have all in succession been used, at the left and right, (as in 99), one additional is expressed by placing 1 at the left of two ciphers; thus, 100; and a farther increase, by repeating with it the preceding combinations (§ 6. 5,) up to 199. Then the figure at the left is increased, and, after that, those at the right; and so on up to 999. 7. For a farther increase, the 1 is removed yet further to the left, and the same combinations are repeated; and so on, indefinitely. †These figures are doubtless called digits, from digitus, a finger, because counting used to be performed on the fingers. § 7. 1. When two or more figures are written by the side of each other, that on the right is called the place of Units, or single ones; that at the left, the place of Tens; the next, the place of Hundreds; the next, the place of Thousands. Then succeed Tens of Thousands, Hundreds of Thousands, Millions, Tens of Millions, Hundreds of Millions; and so on, through Billions, Trillions, Quadrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, &c., &c. 2. Illustration. Hundreds of Quadrillions. Hundreds of Billions. Tens of Billions. - Billions. Hundreds of Millions. Hundreds of Thousands. - Tens of Millions. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3. Explanation. Here 1, in the first place, at the right, signifies one unit, or one; 1, in the second place, one ten, or ten; 1, in the third place, one hundred; 1, in the fourth place, one thousand; 1, in the fifth place, ten thousand; 1, in the sixth place, one hundred thousand; and so on. 4. And such would be the value, or denomination of any figure written in any one of the places. 2 in the place of units, would be read simply 2; 2 in the place of tens, would be read two tens, or twenty; 2 in the place of hundreds, two hundred; and so on. 5. According to this system, the figures increase in value from right to left tenfold. 6. It is known as the decimal system of Notation.* *This system of notation was probably obtained, through the Arabians, from India; and, in the ninth century, took the place of the Sexagesimal system previously used. The remains of this last system we yet have in the divisions of time, where sixty seconds make a minute, sixty minutes an hour, &c. There is no reason in the nature of numbers that their local value should vary according to this law. They might have been made to increase in 3, 4, 5, &c. fold, or in any other. The tenfold increase is assumed because it is most convenient. The circumstance of this increase taking place from right to left is owing to the fact, Decillions. The 7. Three places of figures, commencing at the right, form a period, to which a distinctive name is given. periods are pointed off and named, UNITS, THOUSANDS, MILLIONS, BILLIONS, TRILLIONS, QUADRILLIONS, QUINTILLIONS, SEXTILLIONS, SEPTILLIONS, OCTILLIONS, NONILLIONS, DECILLIONS, &c., &c.t 369, 342, 900, 976, 368, 265, 371, 502, 634, 436. PPPP 9. If no number of one or the other of these denominations is to be expressed, the place of it is supplied by a cipher, in order to preserve to each figure its proper place and value. § 8. 1. CASE I. To write numbers by figures. RULE. Beginning at the left, write by periods, placing each period in its proper order; supplying by ciphers those places and periods that are omitted in the question. 2. Illustration. 41.080.652.941.600.807.362.546.278.009.650.208. that the Arabians and Nations of the East always write from right to left instead of from left to right as we do. †This is according to the French method. places to thousands, six to millions, &c. The English method gives six |