Proposition 18.-Theorem. If a and b be incommensurable, integral values of and y may be found to fulfil the condition at – by=c. Positive integral values of x and y can be found to solve the equation, ax -- by=+1: let these be x1 and y'; then, if we take xo = ct', y = cy', ax! – bye = c, where x and y? are positive integers. A. G. ALGEBRA.—PROBLEMS, continued.— There is a number composed of two figures, of which the figure in the units' place is triple of that in the tens’; and if 36 be added to that number, the sum is expressed by the same digits reversed. Let x be the figure in the tens' place. But 10x + 3.x + 36 = 10 + 3x + 2. ::8=2, and 26 is the number required. Instances of the practical use of equations might be multiplied ; but the intelligent reader will have been convinced of this without the aid of numerous illustrations. With the close of the year, our set of papers on Algebra has ended. Indeed, it would be impossible, within the limits necessarily imposed month by month, to carry the student any higher in the study of this interesting science. To such as are desirous of prosecuting labours thus commenced, Wood's Algebra or find's Algebra may be recommended; and we should advise them to pass on at once to Arithmetical and Geometrical Progression ; afterwards taking Permutations and Combinations, and then proceeding to the Binomial Theorem. When this has been fully accomplished, the student may congratulate himself that his labours, however uninteresting at times, have not been lost, but that, on the other hand, they will prove of inestimable advantage in the study of higher branches of mathematical science, to which, after all, algebra is, in many respects at least, but an introduction. M. L. R. ASTRONOMICAL PHENOMENA. DECEMBER, 1855. By A. GRAHAM, Esq., Markree Observatory, Collooney. The discovery of two small planets, on October 5th, shows that the search for such objects is still kept up vigorously and successfully. Ten have now to be added to the list of twenty-seven, given in the “ Youth's Instructer," for January, 1854. The 28th, in the order of discovery, is due to Dr. Luther, of Bilk, near Dusseldorf, who found it on the evening of March 1st, 1854. Two hours later, Mr. Marsh, in Regent's Park, detected another of the group. These have been named Bellona and Amphitrite. The latter was discovered, independently, in the Oxford Observatory, on March 2d, by Mr. Pogson, and by Mons. Chacornac, on March 3d, in the Imperial Observatory at Paris. Mr. Hind, so justly celebrated, annexed Urania (30) to the catalogue on July 22d. America contributed Euphrosyne (31) on September 1st, through Mr. Ferguson, of Washington, United States. Pomona (32) by Mons. Goldschmidt, Paris, October 26th, is followed on the 28th by Polyhymnia (33), the foundling of Mons. Chacornac, remarkable for the large eccentricity of its orbit : the difference between its perihelion and aphelion distances amounts to twice the mean distance of the Earth from the Sun, or 190 millions of miles. To M. Chacornac we are again indebted for the next in order, Circe (34), discovered 1855, April 6th; and on April 19th Dr. Luther introduces us to Leucothea (35). The two alluded to at the head of this article, forming the 36th and 37th of our littlo sister planets, we owe to the perseverance of M. Goldschmidt and Dr. Luther. On the 31st, half an hour after noon, the Earth will be at its least distance from the Sun, 3,200,000 miles less than its greatest distance, MERCURY will be in aphelion on the morning of the 23d; fifteen millions of miles more distant from the Sun than on November 9th, when he was in perihelion. Here the difference of the distances amounts to more than half the least. This planet will be in descending node on the evening of the 12th ; and on the 31st, in superior conjunction with the Sun, consequently nearly at its greatest distance from the Earth. The angular distance of Venus from the Sun will increase until it reach its maximum value, 47 degrees west, on the evening of the 11th. In this position, a line drawn from the Earth to the planet would be a tangent to her orbit, and hence nearly perpendicular to a line joining the Sun and planet; but the breadth of the illuminated portion of her disc varies as the versed sine of the angle formed by one of these lines with the continuation of the other : hence the planet will at this time be semicircular. MARS will be close to the Moon on the evening of the 2d, and morning of the 31st. Seen from the Earth's centre, the planet would, on the latter occasion, come very close to the edge of our satellite; and a slight parallactic change in the Moon's position, such as will actually take place from many points of view on the Earth's surface, would hide the planet. JUPITER is in Aquarius, twelve degrees southward of the brightest star in that constellation. SATURN will be in opposition with the Sun, at half-past eleven on the evening of the 18th: he, therefore, crosses the meridian on that night almost precisely at apparent midnight, or three minutes before mean midnight. The Sun will have reached his lowest declination, 23° 27' 38" south, at 8h, 48m. on the morning of the 22d. RISING AND SETTING OF THE SUN, FOR THE PARALLELS OF THE BRITISH ISLANDS. Rises. Sets. Rises. Sets. Rises. Sets. Rises. Sets. Rises. Sets. h. m. h. m. h. m. h.m. h. m. b. m. h. m. h. m. h. m. h. m. Dec. 17 36 4 27 45 3 537 55 3 43 8 6 3 32 8 19 3 19 117 48 3 59 7 58 3 49 8 8 3 38 8 21 3 26 8 35 3 12 217 56 4 18 6 3 518 17 3 39 8 30 3 27,8 1 3 12 31 7 59 4 8,8 8 3 58,8 19 3 478 32 3 35 8 46 3 20 SUN AND PLANETS AT GREENWICH. Sun. MERCURY. VENUS. MARS. JUPITER. SATURN. URANUS. b, m. Rises. Sets. Rises. Rises. Rises. Sets. m. h. m. Rises. Sets, h. m. 5 3a, 5 59m. 4 20 5 17 3 38 4 36 2 55 3 56 INDEX TO VOL. XIX. OF THE SECOND SERIES. A Bishop, gold and bronze, 493 Caliph, the word explained, 145 of, 126 Cephalopods, or cuttle-fishes, 166 Charity, disinterested, 505 Christian influence, observations on, 31+ of, 49 sionary, 28-the religions of the 287, 334, 382, 430 of, 529 289 Dudley Stuart, 38—Samuel John- of, by the Rev. H Fish, 300 Devil-u orshippers of Armenia, a visit to the, 498 of, 193 Easter Sunday, rule for finding. 191 Eclipse of the moon, 191, 239, 479-of the sun, 239, 527 306 Franklin, Sir John, Dr. Kane's ex- 190, 239, 287, 334, 382, 430, 479, 527, Free-thinker, a young, on his death- bed, 182 Galileo and his Judges, 150 Good confession of a young convert, sermon by, on the death of a young Greece, history of philosophy in. The fabulous period, 21: Orpheus, 21, tical period, 70: legislators, 70- the peacock, 163—the swallow, 206 sectarian period, 169: Thales, 169- Anaxagoras, Anaximander, and Anaximenes, 245-Socrates, 340 - Plato, 553 209 Happy death of M. 0. I-, of St. New Books continued : Adams's "Greek Text of the Gos- pels, with Prolegomena, Notes, and References," 378 ation," 89 Arago's “Meteorological Essays," 522 Arthur's "Crystal Palace," 196- Landscape Painting in Water- Colours," 136 Bunsen's "Egypt's Place in uni- recent discovery near, 259, 306 — Chapel's "Institutes of Justinian," 507 66-Mohammed's wish to conciliate Copland's" Arithmetic of Fractions Cornucell and Fitch's "Science of Creasy's "History of the Ottoman Farmer's (Miss) “Tonga and the Ferguson's and its Ruins," 89 237, 285, 333, 381, 429, 477, 526, China," 234 Foster's (Mrs. Jonathan) Transla- the Arabs in Spain,"328 Gough's “ New Testament Quota- Crimea," 379 Green's (Mrs. Everett) “Lires of the Princesses of England." 281 of Luther," 329 casus,” 567 and Paper- Howard's (Dean) Translation of the ** Book of Genesis, according to Humphreys's "Manual of Political 18+ Joyce's "Scientific Dialogues, " 475 of the, 191, 239, 479 Kaye's (Bishop) "Government and Discipline of the Church of Christ 474 Kenrick's "Phænicia," 125 " Paper |