One. characters: 6. The ARABIC, or Indian meTHOD is thatwhich is employed in CALCULATIONS,* and is comprised in the use of these TEN 1 7. These characters are, in general, called FIGURES or DIGITS. 8. The first nine are called sIGNIFICANT FIGURES, because they always signify something; and the last NOUGHT or CI PHER, because it signifies nothing. 9. The first figure (1) denotes one single thing or object, and the other significant figures denote in regular succession each one thing or object more than the preceding. 10. If we consider these figures attentively, we shall perceive that no number higher than 9 can be expressed by any ONE of them. They are, therefore, distinguished into ORDERS, and made to signify differently, as they are placed in the 1st, 2d, 3d, &c. order, beginning at the RIGHT HAND, and proceeding towards the LEFT. The next number higher than 9, is called TEN; to denote which we use the same character (1) that we do to represent what is called a simple unit; but to distin guish it from that, we place it on the left hand of a 0; thus 10. The 0 fills the place of units of the FIRST ORDER or of simple units, and the 1 that of units of the SECOND ORDER Or ORDER OF TENS. [It is to be regreted that the names eleven and twelve, (as seen on the following page,) have obtained, for it would certainly be more systematic to call one and ten oneteen, two and ten twoteen, three and ten threeteen or thirteen, &c. It will be seen that there are three names for TEN, viz., ten, teen, and ty; TEEN as in thirteen, fourteen, &c.; TY as in twenty, thirty, &c.] * Derived from the Latin calculare, to calculate. Calculare is derived from calculus, which signifies a small stone; as it is supposed the ancients used to perform their operations by means of small stones or pebbles. †These figures were formerly all called ciphers, hence it came about that the art of arithmetic was called CIPHERING. Derived from the Latin digitus, which signifies a finger; as the ancients used to count by their fingers instead of our counting by these characters. NOTATION. One unit of the THIRD ORDER One unit of the SECOND ORDER called ONE HUNDRED AND ELEVEN. One unit of the FIRST ORDER One unit of the THIRD ORDER One unit of the SECOND ORDER called ONE HUNDRED AND TEN. One unit of the THIRD ORDER, called ONE HUNDRED. No units of the SECOND ORDER, being the PLACE of tens. No units of the FIRST ORDER, being the place of simple units. Here the figure 1 is removed two places to the left, that it may represent TEN TENS, or one hundred. &c. &c. • &c. Three units of the SECOND ORDER, or ORDER OF TENS, called THIRTY. &c. &c. &c. One unit of the SECOND ORDER, or ORDER OF TENS; equal to TEN s. u. One unit of the SECOND ORDER, or ORDER OF TENS, equal to ten s. u. 13,1 2,1 1,1 0 93,2,1 Example 2. Write THREE hundred and FIVE in figures. and three (hundred) represented by 3 in the third place, it will be written the second towards the left, HUNDREDS the third, and so on. five (units) represented by 5, six (tens) represented by 6 in the second place, As, therefore, the number to be written, separated into its parts, contains UNITS always occupy the first place at the right hand, (Art. 10,) TENS In this number are three orders, viz., units, tens, and hundreds. Example I. Write THREE hundred and SIXTY-FIVE in figures. of the second order, TENS; units of the third order, HUNDREDS, &c. Units of the first order, for instance, may be called simply UNITS: units 11. These names may, however, be rendered more concise : 11 1,110,100 thus: there are no tens in this case, we put 0 in the place of tens, and the numthe number will stand: THREE hundred NO tens and FIVE. As, therefore, Here are Two orders expressed, and ONE understood, which, if filled out, ber will be written thus: → Tens & er Units. ∞ Hund. wo Hund. Tens,& er Units. &c. &c. &c. Three units of the FIRST ORDER, or three simple units. Two units of the FIRST ORDER, or two simple units. One unit of the FIRST ORDER, or one simple unit. 11 The 3 in the first place at the right hand, represents 3 units; the same figure in the second place, represents 3 tens or thirty; its value being there increased ten times. Again, the same figure in the third place, represents 3 hundreds, which is ten times the value of it in the place immediately preceding, that is, in the place of tens. This is common to every combination of figures. Hence we have this FUNDAMENTAL LAW in notation, viz., that 13. EACH REMOVAL of place towards the LEFT HAND increases the value of a figure TEN TIMES.* From the foregoing, it plainly appears, that the different values of a figure may be divided into 14. TWO KINDS, which may be called Simple and Local.† I. The SIMPLE VALUE of a figure is what it represents when it stands alone or in unit's place. II. The LOCAL VALUE is determined from the PLACE which it occupies » EXAMPLE. 5 5 5, that is, 5 hundreds 5 tens and 5 (units), or five hundred and fifty five." To write any proposed number in figures, Agreeably to the principles now given, we have this 15. RULE. Write down the FIGURES as their values are expressed, and supply every deficiency in the order with CIPHERS. Example. Write FIVE hundred and Four in figures. Answer. 5 0 (tens) 4, or 504. Write five thousand and four in figures. five thousand and forty. NUMERATION 16. Is the method of numerating and reading numbers. NOTE. For the more easy reading of large numbers, they are usually separated by a comma into half periods of three figures each, as in the following *The reason of this tenfold relation originates in our using TEN figures in our notation. Any other number of figures might be used, but their relation to one an other in combination must always be that indicated by the number of figures. Derived from the Latin locus, which signifies place. NUMERATION TABLE. BILLIONS.f Hundreds of thousands of millions, Hundreds of thousands, Hundreds of millions, cc Tens of thousands, ∞ Tens of millions, Thousands, ∞∞ Hundreds, Tens, -Units, * Two tens and one, i. e., twenty-one. ' Four thousand 321. Fifty-four thousand 321. 21Six hundred and fifty-four thousand 321. ,3 2 1 1 21 4,3 2 1 6 5 321 0,9 8 7,6 5 4,3 2 1 Eighty-seven millions 654 thousand 321. Nine hundred and 87 millions 654 thousand 321. Two hundred and ten thousand 987 millions 654,321. Hence, To numerate any given number, 17. RULE. Numerate from the right hand towards the left, saying units, tens, hundreds, &c. as in the table. And, To READ the SAME, 18. RULE. To the SIMPLE VALUE of each figure, beginning at the left hand, join the name of its place. Numerate and read the population of the city of New-York,.. (1820). . 123,706 | Boston, 43,298 Philadelphia, 108,116 Baltimore, 62,738 Washington, 13,247 New-Orleans, 27,176 State of New-York, 1,372,812 State of Ohio, 581,434 United States of America, 9,636,423 THEORETICAL QUESTIONS. What is Arithmetic? When is it called a science? When an art? What is quantity? When is it said to be discrete? What is number? When is number called abstract? What is a unit, unity or one? When may these terms be applied to any number of ones? What is notation? How many methods of notation are there? What are they? Which is employed in calculations? In what is it comprised? What are these characters, in general, called? What are the first nine called? Why? What the last? Why? What does the first figure denote? What the other significant figures? What appears from the combination of figures ? What is the fundamental Law in notation? How many kinds may the different values of a figure be divided into? What is the simple value of a figure? What the local value? What is the rule by which to write any proposed number? What is numeration? How do you numerate any given number? How do you read the same? * The circumstance of the increase of figures, and consequently of this numerating proceeding from the right hand towards the left, originates in the fact, that this method is borrowed from the Arabic or rather Asiatic nations, who have the habit of writing from the right towards the left. † After Billions proceed Trillions, Quadrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuordecillions, Quindecillions, Sexdecillions, &c. New-York now probably contains near 200 thousand. Numerate and read the following, being all the PERFECT NUMBERS* known. SIMPLE ADDITION. 19. SIMPLE NUMBERS are numbers of the SAME NAME. As 13 dollars, 27 pounds, 36 guineas, &c. 20. SIMPLE ADDITION is finding the sum of two or more six PLE NUMBERS. Illus. 1. Little Edwin has three apples in one pocket, 5 in another, and 7 in his hat; how many apples has he in all? Apples. Here it is plain that we find the answer, for 7 and 5 is 12, and 3 is 15. This separated into its parts, becomes 1 ten and 5 units. The unit figure (5) is placed directly under the row of figures added, and the other figure (1) is removed one place further to the left, that it may represent TEN. 15 Apples. 2. How much money in 3 bags, the first containing 253 dollars, the second 345, and the third 937? Dollars. Answer. Dol. First, we set the numbers down with units under units, tens under tens, &c. Then we begin at the right hand, and add up each row, setting the results underneath as before directed. It will be seen from the operation, that whenever the sum of any row exceeds what can be expressed by a single figure, (Art. 10,) we have to place the figure coming to the left, under the next higher order; and these figures are added in, when adding up the row of that higher order. Hence these two additions may be performed at once: For, we may set down the unit figure of the result of each row, and keep in mind the number of TENS to add in adding up the next left hand row. This adding for the number of tens is called carrying for every ten.† Proceeding in this way, we ob serve the process as expressed in the following language: After setting the numbers down as before directed, we begin at the right hand and add up the row of units, saying 7 and 5 is 12, and 3 is 15; set down 5 and carry 1. . to 3 is 4, and 4 is 8, and 5 is 13; set down to 9 is 10, and 3 is 13, and 2 is 15. Set down the 3 and carry 1 whole sum. * A perfect number is one that is equal to the sum of all its ALIQUOT PARTS. We carry one for every ten, because "each removal of place towards the left hand increases the value of a figure ten times." The reason of this, and indeed of the whole algorithm of arithmetic, originates in the number of figures used in our notation. |