on Transcaucasia; but it contains a clear account of a man and of a tribe who now bear an important part in the struggle against Russia. The History of the Culdees; the ancient Clergy of the British Isles. By the REV. DUNCAN M'CALLUM. (Houlston and Stoneman.) Not indicating his authorities, the author does not enable his clerical brethren to test his accuracy, which, on this particular subject, we should much like to do. But it will interest young people to read of a class of spiritual teachers possessing some features peculiar to themselves. The third and fourth volumes of the Memoirs of James Montgomery, by JOHN HOLLAND and JAMES EVERETT, (Longmans,) are now out. They abound in familiar names. Two other volumes are expected to complete the memoirs, which might have been advantageously compressed into half the space. POETRY. A FRAGMENT. "O that Thou wouldest hide me in the grave, that Thou wouldest keep me until Thy wrath be past." Macclesfield. O HIDE my spirit, hide, Till Thy great wrath be past; Conceal me at Thy side, And be my home at last. Soon as this body dies, To dust returns again, My soul take to the skies: 'Twill be immortal then. Be with us here below, And make our pathway clear; That seeing we may know, Fearing, yet hoping still, Whilst walking by Thy side, THE WANDERER. LESSONS. THEORY OF NUMBERS. Proposition 16.-Theorem. If a and b be incommensurable, each of the numbers a, 2a, 3a,.. (b-1) a, being divided by b, will leave a different remainder. If possible, let two, xa, ya, leave the same remainder, r; that is, let then, xα= mb + r (x − y) a = (mn) b : consequently, (xy) a is a multiple of b; but a is prime to b, by hypothesis; therefore x-y is a multiple of b (Prop. 13); which cannot be, since x, y, and consequently x-y, are each less than 6. This demonstration holds equally whether a or b is the greater. Cor. 1.-The remainders in the proposition embrace all the numbers from 1 to 6-1 inclusive. Cor. 2.-Let xa be that multiple of a which, being divided by b, leaves 1 for a remainder; and let y be the quotient: then, ax = by +1. Hence, the equation ax - by = 1 is possible for integral values of x and y. Cor. 3.-The above equation is identical with, by - ax = - 1. Hence, by interchanging the letters, as in the demonstration a and b were not restricted in their relative magnitudes, ax-by=-1 may be solved by integral values of x and y less than b and a respectively. Proposition 17.-Theorem. If a and b have a common measure m, the condition ax-by=±1 cannot be fulfilled in integers. For, if so, the left side of the equation would be a multiple of m, and not the right. Proposition 18.-Theorem. If a and b be incommensurable, integral values of x and y may be found to fulfil the condition Positive integral values of x and y can be found to solve the equation, ax-by=1: let these be r1 and y1; then, if we take x2= cx1, y2 = cy1, ax2 - by2 = ±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. Then 10x+3x is the number required. But 10x+3x+36= 10 + 3x + 2. .. x = 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 Hind'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 little 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 MARS will be close to the Moon on the evening of the 2d, and JUPITER is in Aquarius, twelve degrees southward of the brightest SATURN will be in opposition with the Sun, at half-past eleven on The Sun will have reached his lowest declination, 23° 27′ 38′′ RISING AND SETTING OF THE SUN, FOR THE PARALLELS OF THE MERCURY. VENUS. MARS. JUPITER. SATURN. URANUS. SUN. Rises. Sets. h. m. h. m. 7 43 3 55 7 55 3 52 8 3 3 53 |