COROLLARY TO (c).—If the terms of a proportion be squared, cubed, etc., or if the 2d, 3d, etc., roots be taken, the powers or roots will be proportional; thus, If 3 29 6, then 33: 28: 98: 68. Why (249, a)? : If 100: 25:: 36: 9, then 100 : /25 :: 36: 9. Why? (d) If two proportions have a common couplet, the remaining couplets will constitute a proportion; for two ratios that are respectively equal to the same ratio are equal to each other; thus, Again, if two analogous terms of one proportion are like two homologous terms of another, then the four remaining terms will be proportional; for, by alternation, the like terms may be made analogous; thus, Let 12:4:15 : 5 ) then by by the above, 15:56 2. 12: 4 : : 15 : 5 12:4:: 6:2 The comparison of proportions having like terms may be varied in many ways. (e) The analogous or homologous terms of a proportion may be increased or diminished by terms having the same ratio (409), without destroying the proportion; thus, if 20: 5 and 12 : 3 have the same ratio as 4: 1, then 20+12:5+3 :: 4:1 and 2012:5 : 4: 1, etc., etc. 3 : 440. The changes that may be made on the terms of a proportion are very numerous, but they are reducible to a few general principles; as, (a) By changing the order of the terms, (b) By multiplying or dividing by the same number, (c) By multiplying or dividing the terms of one proportion by those of another, (c, Cor.) By involving or evolving the terms, (d) By comparing proportions which have like terms, (e) By adding or subtracting terms of equal ratios. REMARK.-A familiar acquaintance with these changes will greatly facilitate the study of Algebra and the Higher Mathematics. 441. A continued proportion is one composed of several ratios, in which the consequent of the 1st ratio is the antecedent of the 2d; the consequent of the 2d, the antecedent of the 3d, etc.; thus, 2:4:: 4:8::8:16::16: 32, etc. 442. In continued proportion the number of different quantities is one greater than the number of couplets and the 1st is to the 3d, as the square of any one of the antecedents is to the square of its consequent; the 1st is to the 4th as the cube of either antecedent to the cube of its consequent, etc. etc.; thus, in the continued proportion given above 2 : 8 :: 22 : 42 or as 42 ; 82, etc, Again, 2: 16: 23: 48 or as 163: 323, etc. 443. Three or four quantities are in harmonical or musical proportion when the first is to the last as the difference between the first two is to the difference between the last two; thus, 20, 16, 12, and 10 are in harmonical proportion, for 20 : 10 :: 20 – 16:12. - 10. 444. If the reciprocals of any arithmetical series of integral numbers be reduced to a common denominator, any three consecutive numerators will be in harmonical proportion; thus, take the series 2, 5, 8, and 11, whose reciprocals are,,, and = - 118, 178, 118, and, and the numerators are in harmonical proportion; for, 440: 110 :: 440-176: 176-110 and 176 80 176-110 110-80. Again, take 9, 7, 5, 3, 1, whose reciprocals are, 4, 1, 3, 1= 945 103, 125, 183, 313, 84, and we have 105: 189 :: 135 — 105: 189135, etc. etc. § 50. THE CALENDAR AND CHRONOLOGICAL PROBLEMS. 445. The only natural and obvious divisions of time are days, months (moons), and years. Other distinctions, such e. g. as hours, weeks, centuries, etc., are artificial, and consequently different nations have made different divisions, and dated their reckoning from different epochs, and thus there has been very great confusion in respect to dates. 446. The Solar Year is the time occupied by the earth in making one revolution in its orbit, as, e. g. in passing from the vernal equinox (the time of equal day and night in the spring) round to that point again. 447. The most ancient nations, by noting the time when a vertical rod, called the stylus, cast the shortest shadow at noon in successive years, discovered that the solar year consisted of 365 entire days; but, by the aid of modern science, the year is found to consist of very nearly 365 days, 5 hours, 48 minutes, and 49.62 seconds. 448. Now, if a year were just 365 days, and the stylus cast the shortest shadow on the 21st of June this year, it would do the same each succeeding year, perpetually; but as a year is nearly 365 days, the shortest shadow, after the lapse of four years, would not be cast until June 22d, and not until June 23d in 4 years more; and thus the summer solstice, (the time of the shortest shadow,) would occur on every successive day in the year. 449. To avoid this confusion, the Roman Emperor, Julius Cæsar, in the year 46 before Christ, introduced a day in February every 4th year, and thus made every fourth year (bissextile or leap year) consist of 366 days; but this correction was too great by more than 11 minutes a year, and consequently in about 129 years the summer solstice would occur one day earlier, say June 20th, and in 129 years more it would occur June 19th, and so on. 450. At the time of the Council of Nice, A. D. 325, the vernal equinox was known to be on the 21st of March; but, by following the rule given by Cæsar, making every 4th year consist of 366 days, the Calendar had been deranged 10 days before the time of Pope Gregory XIII, who, in the year 1582, to restore the equinox to the 21st of March, decreed that the year should be brought forward 10 days, by calling the 5th of October the 15th, and the succeeding days in order, 16th, 17th, etc., and, to prevent similar confusion afterwards he made this RULE. Every year whose number is divisible by 4, except those divisible by 100 and not by 400, shall consist of 366 days and all others of 365 days. 451. A Julian period of 400 years is thus three days longer than the same period by the Gregorian rule; but this is not quite so much as the actual difference between 400 Julian and 400 solar years; for 400 Julian years are 146100 days, while 400 solar years are 146096.896+ days, and 146100146096.896 = 3.1 + Thus even a Gregorian period of 4000 years is a little more than 1 day longer than 4000 solar years. Had Gregory extended his rule by making the years 4000, 8000, etc., to consist of 365 instead of 366 days, as they now do by his rule, the error would be less than 1 day in 100000 years. 452. Dates by the Julian Calendar are in old style (O. S.), and those by the Gregorian are in new style (N. S.). 453. England did not adopt the correction made by Gregory until 1752, when the error in the Julian Calendar was 11 days. Then, by act of Parliament, the year was brought forward 11 days, by calling the 3d of September the 14th, and by the same act the year, which had commenced on the 25th of March, was made to commence on the 1st of January, thus making the year 1751 consist of only about 9 months. 454. In consequence of correcting the calendar, English dates in old style and new, differ from each other not only in the day of the month, but, for that part of the year preceding March 25th, they also differ in the number of the year; e. g. Washington was born Feb. 11, 1731, O. S., but, by new style, the date would have been Feb. 22, 1732, and it is usual, in such cases, to write both years; thus, Feb. 11, 1731–2, O. S.; or thus, Feb. 11, 173. 455. In constructing a calendar, the first problem is to connect the week with the year; i. e. to find the day of the week corresponding with any given day of any year. To do this the first 7 letters of the alphabet are used, A to designate the 1st day of Jan., B, C, D, E, F and G for the 2d, 3d, 4th, 5th, 6th and 7th, and then A is repeated for the 8th and so on through the year. Consequently, one of these 7 letters must stand for Sunday. This letter is called the Sunday Letter or Dominical* Letter; thus, if Jan. begins on Sunday, A is the dominical letter for that year; if Jan. begins on Monday, the 1st Sunday will be the 7th day, and .. G, the 7th letter, will be the dominical letter; ctc., etc. Now, Jan. 1, 1854, was Sunday, and .. A was the dominical letter for 1854; and as a common year consists of 52 weeks and 1 day (= 365 days), 1854 also closed on Sunday; hence, 1855 began on Monday and G was the dominical letter for 1855. Again, 1855 being a common year, closed on Monday, and 1856 began on Tuesday, and .. F was its dominical *Dominical, from the Latin dominus, lord. |