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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.
Then 10x + 3.r is the number required.

But 10x + 3.x + 36 = 10 + 3x + 2.
13.0 + 34 = 30x; ..171 = 34.

::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.



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.



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Rises. Sets. Rises. Rises. Rises. Sets.
h. m. h. m. h.

h. m.
h. m.

h. m.
7 43 3 55) 6 9m. 3 17m. 0 36m. 10 4a.
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8 3 3 531 7 44 3 48 0 12 9 3
8 6 4 0 8 20 4 9 11 56a. 8 35

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A Bishop, gold and bronze, 493 Caliph, the word explained, 145
A caution, for those who need it, 86 Cardinal Wolsey, contemporary sketch
A conclusire argument, 557

of, 126
A death-bed prayer, 566

Cephalopods, or cuttle-fishes, 166
A fact, with a moral, 540

Charity, disinterested, 505
A sister's mission, 99, 155

Christian influence, observations on,
A youthful hero, 258

Academus, at Athens, note on, 554 Chrislian soldiers on the field of battle,
African eloquence, specimen of, 19 128
Amhilion and humility, remarks on, Claude, of Lorraine, biographic sketch

of, 49
America. The young American Mis Comel, De Vico's, re-appearance of,

sionary, 28-the religions of the 287, 334, 382, 430
United States and of South Ame. Conisborough Castle, historic sketch
rica contrasted, 366

of, 529
Anaxagoras, doctrines of, 245 Cook, Captain, biographic sketch of,
" And she was a widow," 423

Anecdotes: Young Victoria, 16-Lord | Cooper, (Mrs. Mary S..) “Memorials"

Dudley Stuart, 38—Samuel John- of, by the Rev. H Fish, 300
son, 87-M. Arago, 214—Anaxi. Correggio, biographic notice of, 97
menes, 377—Rev. S. Kilpin, 540-
Dr. Rush, 557 -Pope Pius VII. and Decision, incitement to, 218
M. Denon, 565

Devil-u orshippers of Armenia, a visit
Anointing, medicinal, 65

to the, 498
Anthony, St., Feast of, in Rome, 308 Drake, Sir Francis, biographic sketch
Arabia, Christianity in, 199

of, 193
Arago, Mons., among Spanish rob-
bers, 214

Easter Sunday, rule for finding. 191
Arctic navigation, 563

Eclipse of the moon, 191, 239, 479-of
Are you converted ? 352

the sun, 239, 527
Armada, Spanish, notice of the, 197 Egypt, Old, slavery in, 34
Ashes of sacrifces, at Jerusalem, 259, Exertion, incitement to, 473

Asteroids discovered during the last Fox, the, described, 261
two years, 571

Franklin, Sir John, Dr. Kane's ex-
Astronomical phenomena, 47, 95, 143, pedition in search of, 563

190, 239, 287, 334, 382, 430, 479, 527, Free-thinker, a young, on his death-

bed, 182
Almosphere, Lieut. Maury's observa-
tions on the, 17

Galileo and his Judges, 150
Aurora borealis, cause of the, 494 Going up the hill, 506

Good confession of a young convert,
Balaclava, described, 63
Bellot, John Frederick, account of, 209 Grampus, round-headed, capture of a
Benson, Rev. S., extracts from a troop of the, 508

sermon by, on the death of a young Greece, history of philosophy in. The
lady, 458

fabulous period, 21: Orpheus, 21,
Bentley, Richard, D.D., biographic 68-other celebrities, 69. The poli.
sketch of, 481

tical period, 70: legislators, 70-
Birds : the her, 81-the cock, 114- the Seven Wise Men, 70. The

the peacock, 163—the swallow, 206 sectarian period, 169: Thales, 169-
.Books, on the choice of, 162

Anaxagoras, Anaximander, and

Anaximenes, 245-Socrates, 340 -
Cain and Abel, 558

Plato, 553


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Happy death of M. 0. I-, of St. New Books continued :
Saviour's, Southwark, 458

Adams's "Greek Text of the Gos-
Hearing without faith, 466

pels, with Prolegomena, Notes,
How to observe, 177, 220

and References," 378
Hunt, late John, reminiscences of, 218 Anderson's "Ladies of the Reforin-

ation," 89
Improve yourselves, 376

Arago's “Meteorological Essays,"
Inkerman, described, 63

Ireland, an old letter about, 464

Arthur's "Crystal Palace," 196-
Irish hospitality, in ancient times, 205 ** People's Day," 281
lean the Terrible, of Russia, latter Barnard', " Theory and Practice of
days of, 416

Landscape Painting in Water-

Colours," 136
Jack and his master, 107

Bunsen's "Egypt's Place in uni-
Jerusalem, mosque of Omar in, 145– versal History," 41

recent discovery near, 259, 306 — Chapel's "Institutes of Justinian,"
notes from, 455

Jewish Sabbath, account of the, 11 Colquhoun': “Sketches of notable
Jews, medicinal anointing among the, Lives, 474

66-Mohammed's wish to conciliate Copland's" Arithmetic of Fractions
the, 251, 251-synagogues of the, and Proportion," 521
in England, 399

Cornucell and Fitch's "Science of
Johnson, Sainuel, and his father, 87 Arithmetic," 184
"Jump in," 280

Creasy's "History of the Ottoman
Jupiter Olympius, the temple of, 385 l'urks," vols. I., II., 41

Farmer's (Miss) “Tonga and the
Kant's theology, character of, 409 Friendly Islands," 135
Kitto, Dr., memoirs of, 344

" Nineveh



Ruins," 89
Lamb, Charles, a caution by, 86 Pielding's Mixed Tints," 522
Lessons: algebra, 46, 94, 142, 189, Fishbourne's Impressions of

237, 285, 333, 381, 429, 477, 526, China," 234
570-eometry, 44, 91, 139, 187, Fleury's " Historical Chapters re-
235, 283, 330, 380---theory of num. lating to many Lands," 41
bers, 427, 476, 524, 569

Foster's (Mrs. Jonathan) Transla-
Lettera, conveyance of, in olden time, tion of Dr. Condé's “ History of

the Arabs in Spain,"328
Lighl, observations on, 563

Gough's “ New Testament Quota-
Lloyd, Henry John, memoir of, 548 tions," 567
Luther, Martin, table-talk of, 134- Grant's "Historical Sketch of the
death and obsequies of, 254

Crimea," 379

Green's (Mrs. Everett) “Lires of
Manners in the olden times, 175

the Princesses of England." 281
Marriage portions, view of Lycurgus Hare's (Archdeacon) " Vindication
respecting, 328

of Luther," 329
Nartyr, Peter, biographic sketch of, Harlhausen's "Tribes of the Cau-

casus,” 567
Masius's “Studies from Nature," Herring's

and Paper-
noticed, 89-extracts from, 81, 114, Making," 474
163, 206, 261

Howard's (Dean) Translation of the
Melancthon, young Professor, 533

** Book of Genesis, according to
Michael Feodorowitch, Czar, biogra- the Version of the LXX.," 473
phic sketch of, 337

Humphreys's "Manual of Political
Mohammed, the Christians, and the Science, for the Use of Schools,"
Jews, 251

Money, observations on, 322

Joyce's "Scientific Dialogues, " 475
Moon, habitability of the, 95-eclipse Kalendar of the Martyrs of 1555,

of the, 191, 239, 479
My library," 404

Kaye's (Bishop) "Government and

Discipline of the Church of Christ
Naturalisl, qualifications of the, 24 during the first three Centuries,"
NEW Books:

A Hand-Book of Anglo Saron Root- Kelly's "History of Russia," 425
Words, 136

Kenrick's "Phænicia," 125

" Paper

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