required the series of fractions converging to the ratio of the circumference and the diameter. In questions like this, in which one term of the fraction or ratio is not precisely given, but is contained between given limits, as when one of the terms is an infinite decimal, it is proper to work for the quotients by both limits, and to use those only which result from both. Thus, in the proposed exercise, by dividing 3.1415926 by 1.0000000, this divisor by the remainder, &c. and by proceeding in like manner with 3.1415927, we find in both cases, the quotients 3, 7, 15, 1, after which the quotients would be different, and are therefore not to be used. Hence the converging fractions are found as in 3 7 15 the margin. It appears, therefore, that the diameter of a circle is to its circumference 1 22 333 355 72 106, 113, nearly as 1 to 3; more nearly, as 7 to 22; more nearly again, as 106 to 333; and still more nearly, as 113 to 355. The degree of approximation of each of these to the given ratio, will be discovered by dividing the first term of each by the second. In this way the last gives 3.14159292, which exceeds the truth by less than a ten-millionth part of the circumference. Had the circumference been taken to a greater number of places, (see note, page 112,) the succeeding fractions would have been found to be, 104348, &c. Exam. 3. Let it be required to approximate the ratio of 141421* to 1. Here the quotients are found to be 1, 2, 2, 2, 2, &c. and consequently the continued fraction the same as in the margin. The converging fractions also are thus found; 1 2 2 2 2 2 1, 1, 7, 11, 11, 98, &c. 1 &c. Ex. 1. The height of mount Etna is 10963 feet, and that of mount Vesuvius 3900 feet; required the approximate ratios of their heights. Answ. 1, 1, 14, 18, 18, 155, 137. 2 The height of mount Hecla is 4900 feet, and that of mount Perdu, the highest of the Pyrenees, 11,283 feet: required the approximate ratios of their heights. Answ. 1, 4, 19, H, 143, &c. 3 Find the approximate ratios of 1 and 3-6055513, (that is, of 1 to the square root of 13.) Answ. §. 4, 4, 1°1, 18, 113%, 137, 23%, (c. 4. Required the series of ratios approaching the ratio of English and Irish acres. Answ. 1, 1, 3, 1, §, 15, 34, 89, 136. 5. The weights of equal bulks of pure water and fluid mercury This is the square root of 2, which therefore is expressed by a unit, with a continued fraction, each of whese denominators is 2 Hence the fraction may be continued with. are as 1 to 13.568; required the series of fractions converging to this ratio. Answ. 13, 14, 24, 75, 502, 357, 1838. 6 It has been computed, that between the years 1696 and 1800, the value of money decreased so much, that in the former year £1 would have procured as much of the necessaries of life as £2 7 11 in the latter. Required the series of fractions approaching to the ratio of the values of money at these periods. Answ. §, 3, 4, Tz, 11%. ON ARITHMETICAL SCALES. THE system of notation for which we are indebted to the Arabians, and which has been employed and exemplified in the preceding pages, proceeds according to the combinations of the number ten, and is such as to correspond to the names given to numerals in almost all languages. The language of every civilized nation, ancient or modern, furnishes names for ten times ten, and for ten imes that product; but none of them furnishes distinct names, of general use, for the powers of seven, eight, twelve, &c.; these, as well as other numbers which are not powers of ten, being denomi nated from a combination of the names of the powers of ten, with the names of units,* when necessary. Thus, for the second power of eleven we have no distinct name, but we call it one hundred and twenty one, that is, the second power of TEN with twice TEN, and one unit: and the third power of fifteen we call three thousand, three hundred, and seventy five; that is, three times the third power of TEN, three times the second power of TEN, seven TENS, and five units. In like manner, in the natural succession of numbers, the first after one hundred, is called one hundred and one, the next one hundred and two, &c., no new names being given, but merely combinations of those previously formed. This remarkable agreement in the numerical language of almost ail nations, seems evidently to arise from the use that is made of the fingers in arithmetical computations, in the ruder periods of society, and very generally by those who have not been instructed in better modes of calculation. Such persons, in counting a number of objects, would naturally distribute them into parcels, each consisting of ten, from the number of the fingers employed in reck out limit, the law of continuation being manifest; fest. The mathematical reader will know, that the number one may be regarded as that power of ten (or indeed of any number) whose index is zero. t oning them: and, if the number of these parcels should be great, it would be natural, in ascertaining their number, to form them into Larger parcels, each containing ten of the smaller; and it is easy to see how this principle would be extended to the numeration of any number of objects however great. It is also evident, that when 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 cbtained 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, sixty, &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 anological 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 lessons 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 adequately and correctly express any number, however great. With the Roman, or even with the Greek notation, on the contrary, we find it impossible to express numbers that exceed a certain magnitude: and, even were additional characters formed to supply this defect, we would find, that calculations, which are performed with great facility and despatch by the decímal notation, would, by either the Greek or Roman system, be excessively tedious and intricate, while the performance of many others 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 of notation 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 classes again 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 proceed still in the same manner, only making the classes consist of two, instead of ten, each, we have the binary scale of notation, in which only the two characters, 1 and 0, are requisite for expressing all numbers: and if the classes be made to consist of three each, we have the ternary scale, in which only three characters, 1, 2, 0, are requisite. In the same manner, it is obvious we may have a quaternary, quinary, duodecimal, trigest mal, sexagesimal, 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, ROOT, or 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. Hence, 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, the expression of 592835 in the decimal scale, will be 2470DH in the duodecimal 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 12)592835 12)49402......11 12)4116......10 12)343......0 12)28......7 2......4 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 * These characters, which may serve the intended purpose as well as any others in the few instances in which they will be employed in what follows, 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 hue 1 and 1, which express eleven. given number is analysed into 2x 125+4× 124 +7 × 128 + 0×122 +10 × 12+11, or 2470DH, according to the notation above adopted. that It will be easily found by proceeding in the same manner, for seven thousand, eight hundred, and fifty four, the expression in the binary scale will 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 denary or decimal, 7854; in the undenary or undecimal, 59D0; in the duodenary or duodecimal, 4666; in the vigesimal, or that 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 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 × 6o+5 × 65+ 0x 6'3 x 63 + 1 × 6o + 4x 6+2, or 226214; which result will be obtained more easily by the operation in the margin. 29 174 6 1047 6 6283 Thus it appears that the decimal scale is only one out of many on the same principle; and, upon due consideration, it will be found not to be the best scale that might be adopted. 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 various works, especially to Professor Leslie's Philosophy of Arithmetic, to Mr. Anderson's Treatise on Arithmetic in the Edinburgh Encyclopædia, and Barlow's Theory of Num 37702 6 226214 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 denominatiosas, |