lulls. Then, as the difference of the errors, if the results be both too great or both too small, or as the sum of the errors, if one result be too great and the other too small, is to the difference of the assumed numbers, so is either error to the correction to be applied to the number which ,produced that error. Note 1. — The rule usually given fails in an important class of questions; but the rule here given, if not the simplest in the resolution of some questions, has the advantage of being applicable in every case. Note 2. — In relation to all questions which in algebra would be resolved by equations of the first degree, the differences between the true and the assumed numbers are proportional to the differences between the result given in the question and the results arising from the assumed numbers. But the principle does not hold exactly in relation to other questions; hence, when applied to them, the above rule, or any other of the kind that can be given, will only produce approximations to the true results. In which case the assumed numbers should be taken as nearly true as possible. Then, to approximate more nearly to the required number, assume for a second operation the number found by the first, and that one of the first two assumptions which was nearer the true answer, or any other number that may appear to be still nearer to it. In this way, by repeating the operation as often as may be necessary, the true results may be approximated to any assigned degree of accuracy. This process is sometimes applied with advantage in extracting the higher roots, when approximate results, differing but slightly from entire correctness, will answer. 2. A and B invested equal sums in trade; A gained a sum equal to £ of his stock, and B lost $ 225; then A's money was double that of B's. What did each invest? Ans. $ 600. 3. A person, being asked the age of each of his sons, replied, that his eldest son was 4 years older than the second, his second 4 years older than the third, his third 4 years older than the fourth, or youngest, and his youngest half the age of the oldest. What was the age of each of his sons? Ans. 12, 16, 20, and 24 years. 4. A gentleman has two horses, and a saddle worth $ 50. Now if the saddle be put on the first horse, it will make his value double that of the second horse; but if it be put on the second, it will make his value triple that of the first. What was the value of each horse? Ans. The first, $ 30; second, $ 40. 5. A gentleman was asked the time of day, and replied, that | of the time past from noon was equal to 38j of the time to midnight. What was the time? Ans. 12 minutes past 3. 6. A and B have the same income. A saves T^ of his, but B, by spending $ 100 per annum more than A, at the end of 10 years finds himself $ 600 in debt. What was their income? Ans. $480. 7. A gentleman hired a laborer for 90 days on these conditions: that for every day he wrought he should receive 60 cents, and for every day he was absent he should forfeit 80 cents. At the expiration of the term he received $ 33. How many days did he work, and how many days was he idle? Ans. He labored 75 days, and was idle 15 days. 8. There is a fish whose head weighs 15 pounds, his tail weighs as much as his head and £ of his body, and his body weighs as much as his head and tail. What is the weight of the fish? Ans. 721b. 9. If 12 oxen eat 3^ acres of grass in 4 weeks, and 21 oxen eat 10 acres in 9 weeks, how many oxen would it require to eat 24 acres in 18 weeks, the grass growing uniformly? Ans. 30 oxen. 10. What number exceeds three times its square root by 11? (Note 2.) Ans. 26.4201048. SCALES OF NOTATION. 589. ° The scale of any system of notation is the law of relation existing between its units of different orders. 590. ° The radix of any scale is the number of units it takes of one order to make a unit of the next higher. Thus, 10 is the radix of the decimal or denary system, 2 of the binary, 3 of the ternary, 4 of the quaternary, 5 of the quinary, 6 of the senary, 7 of the septenary, 8 of the octary, 9 of the nonary, 11 of the undenary or undecimal, 12 of the duodenary or duodecimal, 20 of the vigesimal, 30 of the trigesimal, 6O of the sexagesimal, and 100 of the centesimal. 591. ° In writing any number in a uniform scale, as many distinct characters or symbols arc required as there are units in the radix of the given system. Thus, in the decimal or denary scale 10 characters are required, in the binary scale 2 characters, in the duodenary or duodecimal 12 characters, and so on. In the binary scale use is made of the characters 1 and 0, in the ternary, 1, 2, and 0, &c, the cipher being one of the characters in each scale. In the duodenary scale, eleven characters being required beside the cipher, the first nine may be supplied by the nine digits, the tenth by t, the eleventh by e, and the twelfth by 0. 592i° To change any number expressed in the decimal scale to any other required scale of notation. Ex. 1. Express the common number 75432, in the senary and duodenary scales. Ans. 1341120 and 377e0. Operation. By dividing the given 6)75432 12)75432 number by 6, it is distribut 0,.r^-^r A ,[,>,,,,„, ~ ed into 12572 classes, each 6)12o72 0 12)6286 0 containing 6, with 0 remain 6)20<J5 2 12)523 10, or t der By the second division cYo~iT\ 1 i O\tq 7 by 6, these classes are dis ojd4J i iZ)i6 i tributed into 2095 classes, 6)58 1 3 7 each containing 6 times 6, 6Tg ^ or the second power of 6, <_ with a remainder of 2 of the 1 3 former class, each containing 6. By the third division the classes last found are distributed into 349 classes, each containing 6 of the latter, which were each the second power of 6, and therefore these are the third power of 6, with a remainder 1 time the second power of 6 In like manner, the next quotient expresses 58 times the fourth power of 6, with a remainder 1 time the third power of 6; the next quotient expresses 9 times the fifth power of 6, with a remainder 4 times the fourth power of 6; and the last quotient expresses 1 time the sixth power of 6, with a remainder 3 times the fifth power of 6. Hence, the given number is found to bo equal to 1 X 6* + 3 X 61 + 4 X 6« + 1 X 1 X 6s + 2 X 6 + 0, or according to the senary system of notation 1341120. By proceeding in like manner, we find the given number to be equal to 3 x 124 -\- 7 x 123 -f- 7 X 12s + 10 X 12 + 0, or, according to the duodenary scale, 377e0 Rule. — Divide the given number by the radix of the required scale repeatedly, till the quotient is less than the radix; then the last quotient, with the several remainders in the retrograde order annexed, placing ciphers where there is no remainder, will be the the given number expressed in the required scale. 2. Change 37 from the decimal to the binary scale. Ans. 100101. 3. Reduce 1000000 in the decimal scale to the ternary and also to the nonary. Ans. 1212210202001, and 1783661. 4. How will 476897 in the decimal scale be expressed in the duodecimal scale? Ans. Itee95. 593.° To change any number into the decimal scale, when expressed in any other scale of notation. Ex. 1. Change 377<0 from the duodecimal to the decimal scale. Ans. 75432. • PERATIOl*. 3 7 7<0 l 2 We multiply the left-hand figure by the ra —— dix, and add to the product the next figure: *" then we multiply this sum by the radix, and 1 2 add to the product the next figure, and so 5 2 3 proceed till all the figures have been em 1 2 ployed; and we thus have, as the values of the several figures collected into one sum, 75432, 6 2 8 6 obtained in a manner similar to the reduction 1 2 of compound numbers. Rule. — Multiply the left-hand figure of the given number by the given radix, and to the product add the next figure; then multiply thin sum by the radix, and add to this product the next figure; and so proceed till all the figures of the given number have been added. The result will be the given number in the decimal scale. Note. — When it is required to change a number from a scale other than decimal to another scale also other than decimal, first change the number us given into the decimal scale, and then the result into the required scale. 2. Reduce 234 from the quinary to the decimal scale. Ans. 69. 3. Change 21122 from the ternary to the decimal scale. Ans. 206. 4. Change 100101 in the binary scale to a number in the decimal scale. Ans. 37. 5. Reduce 13579 in the duodecimal scale to the undecimal scale. Ans. 190<3. 6. How will 123454321 in the senary scale be expressed in the duodenary scale? Ans. 9873*1. 594° To perform addition, subtraction, multiplication, division, &c. in a scale of notation whose radix is other than 10, we may Proceed as in the common scale of notation, except that the radix oj .he given scale must be used in the cases wherein the number 10 would be applied in the decimal system. Ex. 1. Required the sum and difference of 45324502 and 25405534 in the senary scale, or scale whose radix is 6. Ans. Sura, 115134440; difference, 15514524. 2. Multiply 2483 by 589 in the undenary scale, or scale whose radix is 11. Ans. 13122^5. 3. Divide 1184323 by 589 in the duodenary scale, whose radix is 12." Ans. 2483. 4. Extract the square root of 11122441 in the senary scale. Ans. 2405. DUODECIMALS. 595. Duodecimals are numbers expressed in a scale whose radix is 12, so that 12 units of each lower order make a unit of the next higher. 596. In finding the contents of surfaces and solids, however, it is customary to apply the term duodecimal to a mixture of the decimal and duodecimal scales. Thus, in admeasurements in which the foot is the leading unit, though the different orders of units are expressed according to the duodecimal scale, the number of units in each order is usually expressed according to the decimal scale. 597. According to this mixed scale, the foot is divided into 12 equal parts, and each of these parts into 12 other equal parts, and so on indefinitely, giving T!2, T|r, &c. In writing these fractions without their denominations, to distinguish their orders, or denominations, accents, called indices, are written on the right of the numerators. Thus, inches are called primes, and are marked '; the next subdivision is called seconds, marked "; the next is thirds, marked "'; and so on. Note. — Numbers expressed by the mixed scale of feet, primes, seconds, &c. may be changed to the pure duodecimal scale, and the operations of addition, subtraction, multiplication, division, and so on, then be performed with them, as in Art. 594, observing to place a point between the unit and its lower duodecimal orders, and in the result changing the figures on the left of the point into the decimal scale, and marking those on the right as primes, seconds, &c, according to their places from the order of units. But the operations of adding, subtracting, &c. are usually performed by other methods, such as are given in the articles that follow. |