For, lets but=" nd =s, then i = 1; «=65, e = dt; a te= (6 d) 1, and =a* bd that is, d b+d6Corollary. In like manner, it , each , &c. Proposition 2.-Theorem. Inte, where is the equivalent fraction whose terms consist of the smallest integral numbers, or, as it is commonly expressed, a fraction in its lowest terms; then a is a multiple of 6, and b of d, and they are equimultiples. For, if not, let c be contained in a n times with a remainder r less than e; that is, let a = nc + r, and let b=nd +$; then, ; *; but a therefore, ne +1_n ne + y - nc (by Proposition 1) =- : nd+$nd nd +$ - nd hence, But r is less than e: therefore, by the nature of fractions, s is less thand, and thus is not the fraction in its lowest terms, contrary to the hypothesis ; c must therefore measure a: but, as often as e is contained in c, d must be contained in b; for, if not, let a = til, and b = nd+s, then, - =*; therefore, ad +5 and nd+s nd contrary to the nature of fractions. Proposition 3.- Problem. To reduce a fraction to its lowest terms. Rule. Divide the numerator and denominator by their greatest common measure; the quotients are the numerator and denominator of the fraction in its lowest terms. Demonstration. Let m be the greatest common measure of a and é, and let e and f be the quotients ; ; is the fraction in its least terms. If not, let be the fraction in its least terms, where c and d e -; but are less than e and f respectively: then, by Proposition 2, a and 6 are equimultiples of c and d. Let a=nc, and b = nd; but a=me, and b=mf, and e and d are less than e and f respectively: therefore n is greater than m; that is, there would be a common measure of a and b greater than the greatest; which is absurd. Proposition 4.-Theorem. If one number measures another, it measures any multiple of that other. Let 6 be contained in a m times, without remainder, it measures na. For a= mb; therefore, na= = mnb, which is a multiple of b. Proposition 5.-Theorem. If one number (c) measures each of two other numbers (a and 6), it measures their sum (a + b), and their difference (a - b). For, if a = me, and b=nc, a+b= (m + n) c; and, a - b = (m - n) C. Proposition 6.-Theorem. If a number (0) measures the divisor (%) and dividend (a), it measures the remainder (r), if such there be. For, since it measures the divisor, it measures every multiple of the divisor (Proposition 4); therefore that multiple which is next less than the dividend : but it measures the dividend (by hypothesis); therefore it measures their difference, which is the remainder. Proposition 7.-Theorem. If a number measures the divisor and remainder, it measures the dividend. For, since it measures the divisor, it measures that multiple of the divisor which is next less than the dividend, and it measures the remainder; therefore, it measures the sum of these two numbers, which is the dividend. A. G. ALGEBRA.- QUADRATIC EQUATIONS, continued.-To prove that in a quadratio of the general form 32 +px+q=0, x=the sum of the two values of x. Let a, ß, be the two values of x; then, a +pa +9=0 ... a? - Be + p (a - b) = 0; and B2 + PB+q=0) i.a + ß + p= 0, and a +ß=-p. To prove that in a quadratic of the general form x + px +q=0, q= the product of the two values of 1. Taking a, B, as the to values, we have in the case of the former a®+pa +9 = 0; or, since p=-(a +ß), .:9= a ß. If, now, we substitute these values of p and g in this general equation, we have 22 - (a +B) + + aß= 0; or, (2 - a) (x-) = 0. So that if the roots of a quadratic equation be given, the equation itself may easily be found. For example, if it be required to find the equation whose roots are 3 and 5, we simply proceed thus, (x – 3) (4 – 5) =0; or, x? - 8x + 15 = 0. These are only one or two of the interesting details presented in the study of quadratic equations : they cannot, however, be enlarged upon within the limits of these papers. Solve the following: - 9-6 2 x*- - 2x V1-1=1}, =} V3. V 2ax *=l, or 등 a + 2ax – SIMULTANEOUS EQUATIONS.—A simultaneous equation is, in point of fact, a compound equation. For example: we cannot discover the values of x and y from the simple equation 2 + y = 13; nor from this other, x + y=5; but if we combine these two, a solation of the difficulty is easily arrived at; thus, zce + 2xy + y = 25; .. subtracting the former, 2xy = 12. We have then, 2 – 2xy + y2 =1, or, *+y=$11. ::= 2 or 3). M. L. R. 9 Or ASTRONOMICAL PHENOMENA. SEPTEMBER, 1855, By A. GRAHAM, Esq., Markree Observatory, Collooney. Toward the middle of the month, De Vico's Comet of short period will be near Aldebaran, the bright star in Taurus, and will rise at about half-past nine in the afternoon. Its increasing distances from the Sun and Earth produce a corresponding diminution of the light. We are ignorant of the precise laws which regulate this diminution, since we are ignorant of the physical constitution of these bodies; but it is usually assumed that the light diminishes as the product of the squares of the distances from the Sun and Earth increases; which is equivalent to the assumption that the apparent light of the Comet is independent of the absolute magnitude of the body, and of the direction from which it is viewed. MERCURY is now receding eastward from the Sun, in the part of his orbit remote from us. The apparent angular distance will be considerable toward the close of the month; yet, so rapid is the planet's southward motion in declination, that his stay above the horizon after the Sun's will, even then, amount to not quite half an hour. On the 15th he will be in the plane of the Earth's orbit going southward. On the 17th he will have the same right ascension as Venus; but will be eight degrees and a half farther north. On the morning of the 26th he will be in aphelion. Venus is approaching the Sun on the eastern side, and will continue to do so till the morning of October 1st, when she will be nearly in a line between the Earth and Sun. Her apparent diameter toward the end of the month will be very nearly one minute of space. With a magnifying power of 300, she would therefore appear under an angle ten times greater than the Moon subtends to the naked eye, and her superficial area would appear a hundred times greater. Her apparent motion in right ascension will be eastward till the 8th ; afterwards westward, or retrograde. The motion in declination will be southward till the 16th. On the 17th she will be at her greatest distance from the Earth's orbit at the southern side. Mars is now in Cancer, near the cluster of stars called Presepe. JUPITER has replaced Venus as the evening star. On the afternoon of the 4th he will be due south at eleven, on the 18th at ten, o'clock. These are the times of his greatest altitude, 25° ; and, consequently, those most favourable for observation. The evenings of the 5th, 12th, 14th, 21st, 23d, and 28th will be richest in phenomena presented by this planet and his satellites. His apparent diameter is 45)'. SATURN rises at hours more seasonable for observation than last month. On the 2d or 3d he attains his greatest northern declination, 22°, 14', 50". He will be near the Moon on the 4th. This planet is approaching the Earth, and thus slightly increasing in apparent magnitude. On the 23d, at three hours in the afternoon, the plane of the equator will pass through the centre of the Sun. Proceeding northward, and retaining its parallelism, with the exception of a very slight nutation, the Sun appears to us to go southward. It is worthy of remark that the time between the vernal and autumnal equinos is one hundred and eighty-six days eleven hours, leaving only one hundred and seventy-eight days nineteen hours from the autumnal to the vernal. During the former period, the Earth passes through the part of its orbit most remote from the Sun: it thus traverses a longer route and with a slower motion than in the latter period.. 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. h. m.sh. m. h, m.in. m. h, m. Sept. 15 13 6 465 10 6 495 6 6 525 2 6 56 4 58 7 1 11 5 28 6 24 5 26 6 265 24 6 28 5 22 6 30 5 19 6 33 215 43 6 25 43 6 2 5 42 6 35 42 6 45 41 6 4 Oct. 15 59 5 405 59 5 39 6 0 5 38 6 1 5 37 6 2 5 36 SUN AND PLANETS AT GREENWICH. SUN. MERCURY. VENUS. MARS. JUPITER. SATURN. URANTS. H. T. & J. Roche, Printers, 25, Hoxton-square, London. |