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GEOMETRY.-Proposition 18.-Theorem. Ip from a point (D), within a triangle (B AC), lines are drawn to the extremities of any side (BC), they are together less than the sum of the other two sides (A B, A C) of the triangle, but contain a greater angle,
Produce one of the lines (BD) to meet a side of the triangle (AC); the other line (DC) is less than the sum of the produced part (DE) and a portion (EC) of that side, by Proposition 17: therefore the two lines (BD, DC) are together less than the sum of the whole produced line (BE) and that portion (EC): but the whole produced line is less than the remaining portion (AE) and its conterminous side (AB); hence, the whole produced line and the former portion (EC) are together less than the two sides (A B, A C) of the triangle; and, a fortiori, the two lines (B D, DC) are less than these two sides.
The angle (BDC) contained by the two lines is external and remote from an angle (BEC) which is itself external and remote from the angle (BAC) contained by the two sides of the triangle : hence, by Proposition 14, the first of these angles is greater than the second, the second than the last; therefore, the first is greater than the last.
By borrowing a little from algebraic notation, the demonstration will be briefer, and more intelligible.
DC<D E+EC, Proposition 17;
hence BD+DC<BA+AC. Again; the angle BDC is external to the triangle DEC; and the angle D EC is external to the triangle BAE: hence
BDC>DEC>BA E or BAC, by Proposition 14. Cor. 1. The straight line (A B) connecting any two points is less than any broken line (AC DE B), consisting of portions of straight lines, connecting these points. In other words, any side of a polygon is less than the sum of all the other sides,
For, by joining alternate angles, beginning with one of the given points, a rectilineal figure (ADB) is formed on the same base (A B), the sum of whose other sides (A D, D B) is less than the broken line, by Proposition 17. By repeating this process, if necessary, we come at last to a triangle on the same base, the sum of whose other sides is less than the broken line; but the base is less than that sum ; therefore, it is less than the broken line.
Cor. 2. If there be two polygons (ACDEB, AFGH KB) on the same base (A B), of which one is internal to the other, and contains no reentrant angle (such as G), the sum of its sides is less than that of the external.
For, if a side (A C) of the internal, conterminous with the base, be produced till it meet a side of the external in c), a polygon (A cGH KB) will be formed whose perimeter is less than that of the external, by Cor. 1. By producing the next side (CD) a polygon is formed, in like manner, whose perimeter is still less. By continuing the process a series of polygons is obtained, the perimeter of each of which is less than that of the preceding : but the external polygon is the first of this series, and the internal the last; hence the corollary.
ALGEBRA.-QUADRATIC EQUATIONS. A quadratic differs from a simple equation inasmuch as it involves the square of the unknown quantity: it is necessary, therefore, to discover from this the value of the quantity itself. Here is an example :
Let x2 + px=9:
4 2. In a quadratic equation the unknown quantity cannot have more than two distinct values. For,
if possible, in the general equation ax? – bx + c = 0, let x bare three different values, a By: then aa® + ba +c=0
(1) aß* + b3 +c=0
(2) aya + by +c=0
Subtracting (2) from (1),
a (a® – B) + b (a - b) = 0
(i.) Subtracting (3) from (1),
a (a- y) + (a - y)= 0,
(ii.) And, subtracting (ü) from (i),
a (B – y) = 0. But a is not* 0, or the equation' would not be a quadratie equation : therefore the factor ß – is equal to 0, or, Bry; that is to say, two of the three supposed values of x are identical, or one in point of fact; and the unknown quantity in a quadratic equation has no more than a double value.,
a'r, I at 9 et
1 a 3 a
b It is a fact that in any equation of the general form of 2* + po +9= 0, - p = the sum of the two values of x; and q = their product. This shall be explained in our next paper : meanwhile, the student will attempt a solution of his own.
M. L. R.
AUGUST, 1855. By A. GRAHAM, Esq., Markree Observatory, Collooney. BEFORB this comes under the eye of the reader, there will probably be more definite information concerning the exact position of the Comet. Still, the predicted places will be found useful in following it, as a small correction, slightly varying, will give the true places.
A comparison of the Ephemeris with a celestial globe or chart shows that the Comet enters Taurus, from Cetus, about the 13th. It rises on August 2d near eleven, September 3d near ten, in the evening.
MERCURY will attain his greatest western elongation, 19° 19', on the afternoon of the 1st. The form will then be semicircular, afterwards gibbous. He will pass from south to north of the plane of the Earth's orbit on the 8th, and will recede from it till the 23d, when his heliocentric latitude, or angular distance at the Sun from the plane of the Earth's orbit, will be 70. On the 13th he will be at his least actual distance from the Sun, on the 26th at his least apparent distance, being, at the latter date, in superior conjunction with our luminary.
The light of VENUS will be at its greatest intensity on the 25th, near the time when the planet is in aphelion. Her distance from the Earth is rapidly diminishing; and, of course, her apparent diameter as rapidly increasing. It is 27 seconds on the 1st of August, and 42 on the 1st of September. Venus will be near the Moon on the 16th.
MARS is now considerably eastward of Saturn. On the 10th he will be within a degree of the star in Gemini marked ; on the 15th, go southward of Castor; and on the 19th, 6° southward of Pollux. Not being very different in apparent brightness from Castor and Pollux, his proximity to these two stars presents a pleasing aspect. The planet will be near the Moon on the morning of the 10th.
JUPITER will cross the meridian twice on the 22d, nearly two minutes after the preceding, and two minutes and a half before the following, midnight. His southern declination being then nearly equal to the Sun's northern, he rises about sunset, and sets about sunrise. He will be easily recognised, pretty low in the south, next in brightness to Venus, though far inferior.
Saturn is now in the Milky Way, on the boundary between Taurus and Gemini, northward of the fine constellation Orion. He will be near the Moon on the morning of the 8th. His apparent diameter is slightly increased since last month; his form is almost the same. The aspects of his eight satellites present in the telescope a constant and pleasing variety, distinguishable, however, only by the highest optical power.
RISING AND SETTING OF THE SUN, FOR THE PARALLELS OF THE
Rises. Sets. Rises. Sets. Rises. Sets. Rises. Sets. Rises. Setx.
b. m. h. m. h. m. h.m. h, m. h. m. h. m. h. m. h. m. b.m. Aug. 14 28 7 44 4 20 7 514 11 8 0 4 2 8 9'3 51 8 2
11 4 42 7 274 36 7 33 4 29 7 404 21 7 48 4 12 7 56 21 4 57 7 8 4 52 7 13 4 47 7 18 4 41 7 24 4 34 7 31 315 12 6 48:5 8 6 515 5 6 555 0 6 594 56 7 4
SUN AND PLANETS AT GREENWICH.
MERCURY. VENUS. MARS. JUPITER. SATURN. URANUS
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