names would be invented, by some process of formation now unknown, for all the numbers so far as ten, and also for the powers of ten, the names of all other numbers would be obtained by a proper combination of these. Thus, the number eleven might be called one and ten; twelve, two and ten; thirteen, three and ten; twenty, two tens; thirty, three tens, &c. and we find this method of denominating numbers strictly followed, except in some of the smaller numbers, such as eleven, twelve, twenty, thirty, &c.; which, from their frequent use, have been more liable to have their names corrupted and altered, but which, when their derivations can be discovered, are always found to be formed on correct, analogical principles, according to the foregoing explanation. Of the advantages and excellence of this system of notation, we can scarcely be duly sensible. Instructed in its use from the earliest notions we receive in arithmetic; never comparing it, or comparing it but slightly, with other modes of expressing numbers by characters; and finding no deficiency, no need of improvement, nothing to call our thoughts to the subject, we use it without feeling its superiority, and with a very inadequate idea of its power. We do not reflect, that merely by means of the different positions and combinations of no more than ten simple characters, we can correctly and easily express any number, however great. With the Roman, or even with the Greek notation, on the contrary, we cannot express numbers that exceed a certain magnitude: and, even were additional characters or marks formed to supply this defect, we should find that calculations, which are performed with great facility and despatch by the decimal notation, would, by either the Greek or Roman system, be excessively tedious and intricate, while the performance of many would be almost impracticable. * But, though the decimal system of notation has so far the advantage over these, and all other systems not depending on the same principle, we must not conclude that no other system of equal excellence could be invented. There may be an indefinite number of systems, founded on the same principle, and possessing various degrees of excellence. In the decimal notation, we distribute numbers into classes or parcels of ten each; these into higher classes, each containing ten of the lower; these into still higher classes, each containing ten classes of the second order; and so on, till the numbers are exhausted. But if we proceeded still on the same principle, only making the classes consist of two, instead of ten each, we should have what is called the binary scale of notation, in which only the two characters, 1 and 0, are requisite for expressing all numbers and if the classes were made to consist of three each, we should have the ternary scale, in which only three characters, 1, 2, 0, are employed. In the same manner, it is obvious, we might have a quaternary, quinary, duodecimal, trigesimal, sexagesimal, *If the learner should attempt, for instance, by means of the Roman notation, to square 1728 (M.DCC.XXVIII.), which is effected in a few seconds by the modern process, he would have some idea of the immense superiority of the one system above the other. centesimal, or any other scale, by merely taking 4, 5, 12, 30, 60, 100, or any other assigned number, as the number contained in each class. This number may be called the RADIX, the ROOT, or the BASE, of the system; and it is obvious, that in each system there will be as many distinct characters required as there are units in the radix. Thus, in the decimal scale, ten characters are necessary, but in the duodecimal twelve would be required, which number would be made up by adding to the characters at present in use two others to denote ten and eleven. In what follows, D will be used to denote ten, and н to denote eleven; * and, in the duodecimal scale, twelve will of course be written 10. 12)592835 From these principles, we have, obviously, the following rule:To express a given number in any assigned scale: Divide the given number by the radix of the scale; divide the result also by the radix, and the result arising from this again by the radix. Continue the division in this manner as long as possible, and to the final quotient annex the several remainders in a retrograde order, placing ciphers where there is no remainder. Thus, 592835 in the decimal scale, will be expressed by 2470DH in the duo .11, or H .10, or D 12)343.. ..0 12)28.. ..7 2......4 decimal scale, as will appear from the annexed operation. Here it is evident, that by dividing the given number by 12, it is distributed into 49402 classes, each containing 12, with the remainder 11. By the second division by 12, these classes are distributed into 4116 classes, each containing 12 times 12, or the second power of 12, with a remainder of 10 of the former classes, each containing 12. By the third operation, the classes last found are distributed into 343 classes, each containing 12 of the latter, which were each the second power of 12; and therefore these are each the third power of 12; and the remainder is 0. In like manner, the next quotient expresses 28 times the fourth power of 12, and the remainder, 7 times the third power of 12; and the final quotient expresses twice the fifth power of 12, with a remainder of 4 times the fourth power of 12. Hence, the given number is analysed into 2 × 125 + 4 × 124 + 7 × 123 + 0 × 122 + 10 × 12 +11, or 2470DH, according to the notation above adopted. It would be easily found by proceeding in the same manner, that for seven thousand, eight hundred, and fifty-four, the expression in the binary scale would be 1111010101110; in the ternary, 101202220; in the quaternary, 1322232; in the quinary, 222404; in the senary, 100210; in the septenary, 31620; in the octary, 17256; in the nonary, 11686; in the undenary, or undecimal, 59d0; in the duodenary or duodecimal, 4666; in the vigesimal, or that These characters, which may serve the intended purpose as well as any others in the few instances in which they will here be employed, may be easily recollected by conceiving the first to be formed by running together 1 and 0, the characters which express 10 in the common notation, and by joining with a line 1 and 1, which express eleven. whose radix is twenty, (19)(12)(14); in the trigesimal (radix thirty), 8(21)(24); in the quinquagesimal (radix fifty), 374; in the sexagesimal (radix sixty), 2D(54); and in the centesimal (radix one hundred), (78)(54): where each pair of the figures enclosed in brackets would be represented by a single character, were there a sufficient number of distinct characters for each scale. 4503142 6 29 The converse of this problem, or the reduction of a number to the decimal scale from any other, will be performed by finding the values of the several digits, and collecting those values into one sum; or, more easily, by multiplying the left-hand digit by the radix, and adding to the product the next digit; then by multiplying this sum by the radix, and adding to the product the next digit, and so on, till all the digits shall have been employed.* Thus, 4503142 in the senary scale, is equivalent to 4 x 66 + 5 × 65 + 0 × 6 + 3 × 63 + 1 × 62+ 4 ×6+2, or 226214; which result will be obtained more easily by the operation in the margin. It does not suit the plan or the limits of the present work, to add much to what has been said on this curious and interesting subject. For farther information, recourse may be had to Barlow's "Theory of Numbers," or to any good modern treatise on algebra. 6 174 6 1047 6 6283 6 37702 6 226214 With regard to different scales, it may be sufficient to observe, that the binary is chiefly important from its unfolding some curious properties of numbers; that from its employing no character of higher value than unity, operations would be performed by it, though tediously, yet with great facility, and with little mental labour; but an insuperable obstacle to the general use of this scale, is, that in expressing numbers, the characters 1 and 0, must generally be repeated a great number of times. The same advantages, and the same disadvantages belong, but in a less degree, to the ternary scale, and still less to the quaternary. The quinary, septenary, and undenary all labour under the disadvantage of having their bases prime numbers, and of thus giving origin to a great number of interminate fractions.† The octary and nonary scales are not so objec *It is scarcely necessary to remark, that both this rule and the preceding are the same in principle, as the rules for the reduction of quantities of different denominations. †That is, interminate fractions having for their denominators the radix of the scale, or its powers, such as periodical decimals in the common scale. The numbers which, when used as denominators in the common notation, do not give origin to interminate decimals, are the powers of 2 and 5, and the products of those powers; such as 2, 4, 8, &c. ; 5, 25, &c.; 10, 20, 40, &c.; 50, 100, &c. In the senary and duodenary scales, the corresponding denominators are the powers of 2 and 3, and the products of those powers; such as 2, 4, 8, &c. ; 3, 9, 27, &c.; 6, 12, 24, &c.; 18, 36, 72, &c. Now, it is easy to show, that, in any considerable interval, there will be almost half as many more numbers of this kind in either of the latter scales, as there are in the decimal scale, the ratio tionable on this account, but they have no advantage of a different kind to recommend them. The decimal scale does not give origin to so many interminate fractions as either of the latter, and has besides the advantage of expressing numbers rather more concisely. The senary and duodenary scales, having each so many integral aliquot parts in proportion to its magnitude, and those of so convenient a kind, give origin to much fewer interminate fractions, than any of the above-mentioned. These two are preferable, therefore, in a considerable degree, to any of the others that have been mentioned. The duodecimal has the advantage of expressing numbers concisely, saving one figure in fourteen or fifteen, as compared with the decimal scale; while a number expressed by four figures in the decimal scale, will ordinarily require five in the senary. This slight want of conciseness, however, in the latter, is perhaps more than counterbalanced by the greater facility with which operations would be performed by its means. The largest character employed in it would be 5; while, in the duodecimal scale, all that are used in the decimal one would be required, and two additional ones for ten and eleven and hence it is easy to see how much more laborious the operations would be in the latter, and how much greater the chance of committing errors. To introduce either of these scales now, however, when men are accustomed to the decimal scale; when the languages of all civilized nations are suited to it; and when so many valuable works, particularly tables, in which it is adopted, would be rendered comparatively useless, would be unadvisable, and perhaps impracticable: but we must regret that the decimal scale was adopted at a time, when any other might have been introduced with equal facility. The following exercises are annexed for the use of those who may wish to familiarize themselves with this subject: Exer. 1. Reduce 1,000,000 in the decimal scale, to the ternary scale, and also to the nonary. Answ. 1,212,210,202,001, and 1,783,661. 2. Reduce 1,000,000 in the quaternary scale, to the denary and binary scales. Answ. 4096; and 1,000,000,000,000. 3. Reduce 123,454,321 in the senary scale, to the duodenary scale. Answ. 987,3d1. 4. How will 476,897 in the decimal scale, be expressed in the duodecimal scale? Answ. 1DH,н95. 5. How will 6666 in the decimal scale, be expressed in the binary and quinary scales? Answ. 1,101,000,001,010; and 203,131. 6. Reduce 13579 in the duodecimal scale, to the undecimal scale. Answ. 190D3. being that of the logarithms of 5 and 3. Thus, under 100 there are thirteen such numbers in the decimal scale, and nineteen in the senary or duodenary; while, under 1000, there are twenty-seven in the one scale, and thirty-nine in each of the other two. When the radix is a prime number, the only denominators of the kind which we are considering, are the powers of the radix. Hence, in the septenary scale, there will be only three such denominators, 7, 49, and 343, below 1000, and in the undenary only two, 11 and 121. QUESTIONS, WITH THEIR SOLUTIONS.* 1. What number taken from the square of 48, will leave 16 times 54? Solution. 482-2304, and 54 × 16-864; then 2304-864 = 1440, the answer. 2. Divide £1000 among A, B, and C, and give A £120 more than C, and C £95 more than B. Sol. A's share is evidently to be 215 ( = 120 +95) more than B's; therefore 1000-215-95690; and 690 ÷ 3 = £230 = B's share. Hence, 230 +95 = £325, C's share; and 325 + 120 = £445, A's share. 3. A father left to his eldest son of his property, to his second of the remainder, and to his third son what was left. What was the share of each, the shares of the first and second differing by £500? 79 = 1225, the part of the whole pro Then,請 Sol. 1-4435, and 35 of 25 perty belonging to the second son. 1-1701-154, the share of the youngest. 6241 6241 + 1225 4701 and Also 44 1225 2251 = 79 6241 6241 the difference of the shares of the first and second. Then, as 2251 6241 :14, or as 2251: 3476:: £500 £772223, the share of the eldest, so is 2251 1225: £500 £272,228, the share of the second and is 2251 1540::£500: £3422581, the share of the third. 4. A gets £4 of a legacy for £3 that B gets, and C £5 for £6 that B gets, and A's share is £5000. What is the whole legacy? Sol. As £3: £6:: £4: £8, A's part, when B gets £6, and C £5. Then as 8 : 8+6+5::£5000: £11,875, the whole legacy. 5. A person possessed of of a ship, sold of his share for £1260. What is the value of the whole ship at the same rate? Sol. of; and as 1:1::£1260: £5040, the answer. 6. A person being asked the hour of the day, said, that the time past noon was of the time till midnight. What was the hour? Sol. As 1+, or as 9: 4::12 hours: 5 hours, 20 minutes; so that the time was 20 minutes past 5 o'clock in the afternoon. 7. A, B, and C purchase a ship; A pays 2, B, and C £2000, of the cost. What are the sums paid by A and B? 32 32 Sol. +3, and 1-33-33, C's part. Then, as 31 14:: £2000 £903; the part paid by A; and as 31 18:: £2000 £1161, the part paid by B. *These questions are selected from various authors. Their solutions are annexed for the purpose of showing the more advanced student, how they and similar arithmetical questions may be resolved. They will be found useful in preparing the pupil for working the Miscellaneous Questions that follow them, and for resolving any of the more difficult questions that may be proposed for solution by common arithmetic. |