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Are shining yet.
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Each bitter smart !
Can always prove
Rule. Find the least common multiple of the first two, next of this and the third, then of this last and the fourth, and so on to the last. Demonstration.
Let M, M, N, be the successive resultery common multiples, and a, b, c, d, &c., the numbers. Whatever number is a multiple of a, b, c, is a multiple of M (Proposition 9); therefore it is a common multiple of M and ci consequently, the least common multiple, M, of Mand c, is the least common multiple of a, b, and e. The same reasoning applies to each successive additional number.
Proposition 13. If a number (c) measures the produet of two numbers (a, b), and be prime to one of them (6), it measures the other (a).
By the process of finding the greatest common measure we have, if c be less than 6,
c) 6 (2
e) d ($
&c. The last divisor i = 1, since c is prime to b.
ab= qac + ad. But ab and qac are multiples of c; therefore ad is a multiple of c.
ac = rad + ae : ac and rad are multiples of c; therefore ae is a multiple of e. By continuing the process we find, ultimately, that al= a is a multiple
The same result would be obtained by a similar method if I be less than c.
Proposition 14. If a number (0) be prime to each of two numbers (a, ), it is prime to their product (ab).
If not, let p and ab have a common measure m, which must, of course, be prime to a and b, since p is.
m measures ab, and is prime to b; therefore it measures a (Proposition 13), contrary to the hypothesis.
Proposition 15. If a series of numbers (P, q, r, &c.) be each prime to each of another series (a, b, c, &c.), the product of the terms of the first series is prime to that of the other.
p is prime to a and b; therefore to ab.
to ab and c; therefore to abc. In like manner, 9, +, &c., are prime to abc. abc is prime to p and q; therefore to pq.
to pq and r; therefore to pqr. Cor. 1.--Hence, if two numbers be prime to each other, any positive integral power of one is prime to any positive integral power of the other.
Cor. 2. -Any power of an irreducible fraction is an irreducible fraction.
Cor. 3.- No root of an integer is an irreducible fraction. Hence, if an integer have not an integral root, its root can only be approximately found.
ALGEBRA.—PROBLEMS PRODUCING SIMPLE AND QUADRATIC EQUATIONS.-One of the most immediate and practical uses of equations is that of solving problems which involve some unknown quantity. A few examples will show their importance in this respect.
1. At what moment between the hours of 4 and 6 will the two hands of a clock be coincident?
At 4 o'clock the minute-hand is at XII. on the dial, whilst the hour-hand is at IV., thus having a start of 20 minute-spaces; and, as the minute-hand travels twelve times as fast as the hour-hand, for every space, ., traversed by the latter, we must allow 12% for the former. The following equation, therefore, naturally occurs to us :
20+4 = 12x;
14 minutes, or 1 minute 49,
41 seconds, by the dial, is the space traversed by the hour-hand before overtaken by the minute-hand. In other words, the two hands are coincident at 21m. 49, s. after 4. This may be verified on any well-constructed clock.
2. Find a number, such, that whether it is divided into two or three equal parts, the continued product of the parts shall be the same. Let x be that number. Then * = * *
:: 43 = 27, or 2 = 63.
3. There is a certain number, of which the cube root is one-fifth of the square root. Find it.-Answer, 15625.
4. A constable in pursuit of a thief finds by inquiry that the thief is travelling l} miles per hour quicker than himself; he therefore doubles his speed after the first 4 hours, and overtakes the thief at the end of 6 hours and 20 minutes from starting. Now the thief had an hour's start, and never varied his speed. At what distance was he
captured; and what had been the rates of travelling of pursuer and
Constable's speed at first, 81 miles per hour.
M. L. R.
NOVEMBER, 1855. By A. GRAHAM, Esq., Markree Observatory, Collooney. As in May the total eclipse of the Moon was followed by a partial eclipse of the Sun, so now the lunar eclipse of October 25th is followed by a partial solar eclipse on November 9th, visible in Tasmania, New-Zealand, and the regions round the south pole. The Moon's disc will not intercept more than half the Sun's diameter, even where the obscuration is greatest. The penumbral cone will first touch the surface of the Earth at 5h. 35m. in the afternoon, at a point about three hundred and fifty miles north of New Zealand; the last contact will be at 9h., within the Antarctic Circle, near the meridian of Greenwich.
MERCURY will be in inferior conjunction with the Sun a little after noon on the 3d, about half a degree southward of the Sun's disc. He will afterwards recede westward till the 20th, the time of greatest elongation, when the angular distance of these two bodies will be nearly 20 degrees. The apparent motion of the planet will be north-westward till the 12th, after the station south-eastward. On the 4th he will be in the plane of the Earth's orbit; on the 19th, at the greatest distance northward from it. About the 21st he will rise nearly two hours before the Sun, and may thus, with a favourable atmosphere, be easily seen by the naked eye.
VENUS will have attained her greatest brilliancy on the morning of the 6th. She will suffer somewhat by the proximity of the Moon, then only 4 or 5 degrees distant; but the contrast will be exceedingly pleasing, even to a casual observer. In a telescope magnifying only fifty times, the planet will appear as large as does the Moon to the naked eye: in fact, both can be readily examined in this way at the same time, and thus the magnitudes directly compared. We recommend those who are furnished with a telescope to make the experiment: it will give a very clear idea of the magnifying power of the instrument. Venus will be in her ascending node on the 12th.
MARS may be distinguished about 25 degrees farther westward than Venus. On the morning of the 4th he will be close to the Moon. It would require a magnifying power of four hundred, to make his disc appear as large as hers. At the end of the month he will be at his greatest distance from the plane of the Earth's orbit.
JUPITER will be close to the Moon on the evening of the 16th, his apparent diameter nearly equal to that of Venus on the 6th; 60 that a similar comparison may be made with our satellite. He is receding from the Earth, and approaching the Sun.
SATURN is still in the Milky Way, about 4 degrees westward of the third-magnitude star, i Geminorum. His apparent motion among the fixed stars is slowly retrograde. The diameter of the globe continues about 2 seconds less than the exterior minor axis of the ring, which is 20 seconds; the major axis is 46 seconds. On the 20th the area of the planet's disc is exactly one-fourth that of Jupiter. RISING AND SETTING OF THE SUN, FOR THE PARALLELS OF THE
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. Nov. 16 49 4 386 54 4 336 59 4 277 6 4 21 7 13 4 14
117 5 4 22 7 12 4 16 7 19 4 9 7 27 4 17 36 3 51
217 21 4 10 7 29 4 2 7 38 3 54 7 48 3 447 59 3 33 Dec. 17 36 4 2 7 45 3 63 7 55 3 43 8 6 3 32 8 19 3 19
SUN AND PLANETS AT GREENWICH.
MERCURY. VENUS. MARS. JUPITER. SATURN. URANUS.
Rises. Sets. Rises.
Rises. Rises. Sets.
6 40 5 59
PHASES OF THE MOON.
9th day, 7h, 31m. aftern.
23d day, 7h. 5lm. aftern.
H. T. & J. Roche, Printers, 25, Hoxtor-square, London.