In the same manner it may be shown that the sum of the first four Subtrahends is equal to the cube of the first four root-figures; but, in that demonstration, t will represent 946 (tens), and u will stand for the fourth root-figure (3). Now, the four Subtrahends are together equal to the given number 847396215847, for after subtracting them from it, nothing remains. But they are also .. 847396215847 .. 9463 cube root of 847396215847· To extract the Cube Root of any Number. First. Point the number into periods by placing a dot over the units' place, and also one over every third place to the right and left of units. Second. Put under first period the greatest cube num-. ber less than it, and put cube root of that number for first root-figure. Subtract; and to right of remainder bring down next period, making First Dividend. Third. Put root-figure in left-hand column. Multiply by itself, and by 3, carrying out product in right-hand column. This is First Trial Divisor. Fourth. Ascertain how often Trial Divisor is contained in Dividend. Quotient is new root-figure. Fifth. Over the former root-figure standing in lefthand column, put twice that figure. Add the two lines. Sixth. To the sum annex new root-figure. Multiply by new root-figure. Carry out product in right-hand column, under Trial Divisor, but two places to the right. Add with Trial Divisor. The sum is First Complete Divisor. Multiply this by new root-figure. Carry out product (which is Second Subtrahend) under First Dividend. Subtract. Take down next period, making Second Dividend. Seventh. Under Complete Divisor put square of lastfound root-figure. Add with Complete Divisor, and with the number above it (three lines). This makes new Trial Divisor, with which find new root-figure (as in Fourth Step). Eighth. Add twice last-found root-figure to the number in left-hand column. Repeat Sixth, Seventh, and Eighth steps as often as required to complete the example. NOTE I. When a cipher occurs in the root, annex one cipher to number in left-hand column, and two ciphers to Trial Divisor. Take down next period, &c. NOTE II. To find the cube root of a Vulgar Fraction, take those of its Numerator and Denominator. 11. Find /99425242966-036381696 12. Extract 3/31418376777267496615936. THE FOURTH, SIXTH, AND OTHER ROOTS. Any root whose exponent is a power of 2 or 3, or the product of any of their powers, may be extracted by means of the methods explained above. Thus : 4th root = square root of square root. 9th root = EXERCISE CXXXIII. (continued.) 13 Extract the fourth, or biquadrate, root of 9971220736⚫ 14. Find the sixth root of 216270112515625. 15. What is the ninth root of 31418376777267496615936. 16. Extract, to seven places each, /3 and 2/48. 17. How long is the edge of a cube, the solidity of which is 112* cu. ft. 10" 11" 4""? 18. A cubical tank will contain 23 tons 18' cwt. 1. qr. 21. lb. 15 oz. of water, sp. gr. 1 How much ground does it stand on, the material being 3 inches thick? 19. A certain square prism is 4 ft. 6 in. long, by 4. ft. wide and 1. ft. 6 in. thick. Required the edge of an equal cube. 20. What is the whole superficial area of a cube whose solidity is 496 cu. ft. 1" 11" 11" ? 21. How long is the internal edge of a cubical vessel which will contain 16 hhds. 31 gals. 3. qts. 3. gills? APPENDIX. Note to Page 175. OLD AND NEW STYLES OF RECKONING TIME. THE JULIAN AND GREGORIAN CALENDARS. The necessity for two years of different lengths, the Common Year of 365 days, and the Bissextile or Leap Year of 366', arises from the fact that the earth does not make an exact number of diurnal revolutions on her axis during her annual circuit round the sun. Hence the Tropical Year is not an exact multiple of the Solar Day, but contains 365. such days, and a fraction (365-24224 days). Down to the time of JULIUS CESAR, no practical notice of this fraction was taken in the computation of years, and inextricable confusion of dates was the result. For, taking the fraction at exactly of a day, the neglect of it would make every fourth year's date expire one day too soon; one day, that is, before the earth had completed her fourth annual journey round the sun. And, as 365 X41460', in that number of dated years one whole such year would be gained. For 1460 such years would only represent 1459 complete revolutions of the earth about the sun; and, hence, the year which would be called 1460 after any given epoch, would be, in reality, only 1459 years after it. The seasons, too, would, by this time, have made the entire round of the year. The longest day, for example, would have fallen, in succession, on every date in the year. These considerations shew the serious nature of the error which Julius Cæsar sought to remedy, by the addition of an intercalary day (Feb. 29th) to every fourth year; a contrivance for retarding dates that would have been perfectly effectual if their rate of acceleration had been exactly one-fourth of a day per annum. They had not, however, been advancing so rapidly as this; for the Tropical Year is less than 365-25 days, by 00776 of a day, amounting to 6 984 days in 900 Julian Years. The Julian scheme, therefore, over-corrected the error which it was intended to counteract, and retarded the advance of dates, more than was necessary, by nearly 7 days in 900 years. This new error had produced such inconvenience by the time of POPE GREGORY XIII., that, in 1582, he promulgated, for its correction, the rule which bears his name. It is that which is explained on page 175, and it prescribes that, in every 400. years, three which, under the Julian scheme, would have been leap years of 366 days each, shall be common years of 365 days each. This plan accelerates the Julian speed of dates by 6.75 days in 900 years, and, thus, so very nearly counteracts the error, that 10,000 Gregorian Years differ by only 2-6 days from the like number of Tropical Years. And this error might be reduced to a single day in 100,000 years, by (extending the Gregorian principle one step further) declaring that every year divisible by 4000 shall be a common year of 365 days, as suggested by Sir John Herschel. To 365 days X 100,000. Add 1 day for every 4th year PROOF. .. 100,000 JULIAN YEARS contain == When the Gregorian Calendar was promulgated, in 1582, the accumulated error of the Julian computation amounted to 10 days, which number it was, consequently, necessary to strike out, and this was done by calling the next day to October 4th, October 15th, instead of October 5th. The Gregorian Calendar was not adopted in England until 1752, when 11 nominal days were cancelled, by placing the 14th of September next to the 2nd of that month. And these two days are, respectively, the first of New Style, and the last of Old Style, in English Chronology. Russia is the only European country in which O.S. is still retained. There is now a difference of twelve days between the two Styles, so that a Russian's New Year's Day is our January 13th. Note to Pages 87 and 207. When borrowing is necessary, Subtraction is always effected by the aid of Addition. For, we say either, 1. Borrowed Quantity + Minuend-figure Subtrahend figure. or, II. Borrowed Quantity - Subtrahend-figure + Minuend-figure. But, we might dispense with the Addition, by saying, III. Borrowed Quantity — (Subtrahend-figure — Minuend-figure). EXAMPLE OF PURE SUBTRACTION. From 3056002143. 98153687. Take 2957848456. as follows: Here, though we cannot take 7. from 3. we can take 3 of the 7 from the 3'; leaving only 4 to be taken from the borrowed ten. The Work, without employing Addition, is 3 from 7 four; 4 from ten 6 3 from 8 five; 5 from ten 5 0 from 6 six; 6 from ten 4 CHRISTIE'S CONSTRUCTIVE ETYMOLOGICAL SPELLING-BOOK. CRITICAL NOTICES. Athenæum. "An elementary spelling-book, on a partly new and very satisfactory plan, as it seems to us, the leading features of which may be thus briefly explained:-The book begins with the first alphabetical prefixes A, AB, or ABS (Latin), signifying from or away; A, (Anglo-Saxon) on, in, to, at, and those words are then given, with their meanings, to which these are attached. The other prefixes and affixes follow in their proper order. Thus, the learner, from the common germ, gets a clue to the sense of a whole family of verbal variations. In a second part of the work, the Greek or Latin root is placed at the head of the column, with the radical meaning attached; and underneath, the list of words into the composition of which it enters;-e.g. Agon (Greek), a struggle; agony, intense pain; agonize, to strive painfully; antagonist, an opponent; antagonism, opposition. There are also interspersed throughout the volume a number of foot notes, explaining such little difficulties as are likely to puzzle the learner." Educational Record. "The title of this little book sufficiently expresses its purpose. Its author has had much experience and success as a teacher, and he has here embodied the result, not only of his experience, but of much research and labour in a very important field. The study of the origin and derivation of English words has become an indispensable part of even elementary education, and we do not know how it can be better forwarded than by the adoption of this sensible and well-arranged volume." Papers for the Schoolmaster. "We have been in schools where the teachers have most miserably failed in attempting etymology, and this too, where there was really nothing to gain by tracing the word back to its root; and in many cases where it was well done, we were not at all satisfied that the result was worth the pains. Both for teachers and their apprentices, we do not know a Manual of Etymology that would serve them better than the one now under notice." The Critic. "This little work, though written for elementary schools, contains much valuable matter upon the derivation of the principal words in our language. The compiler has bestowed great care and labour on his task. As possessing a vast superiority over the numerous spellingbooks put into the hands of children, it must ensure an extensive use. Educational Times. "We have no doubt this little work will prove very useful. The arrangement is made with considerable intelligence, on a plan developed to a degree we have not yet met with, and bearing every evidence of greater usefulness than those already employed." |