seems to possess at least the second characteristic of an axiom ; for, notwithstanding the repeated efforts of geometers, it is not certain that it has ever been deduced from truths more elementary by reasoning intelligible to ordinary minds on the threshold of science. Postulates.-1. Let it be granted that a right line may be drawn from any one point to any other point. 2. That a terminated right line may be produced to any length in a right line. 3. And that a circle may be described from any given point as centre, with a radius equal to any given finite right line. The first and second postulates concede the use of the straight-edge for drawing right lines; the third, the use of the compass for describing circles. No solution of a problem is admitted in elementary geometry which has not been effected by means of the operations here indicated. In what follows, we have not wantonly departed from Euclid's arrangements. The desire to present the more useful elementary truths of Geometry in a small compass, has caused the omission of some propositions which are not necessary for the chain of reasoning. Those which have been added, will be found useful in other branches of mathematics. We have endeavoured, to make the solutions of the problems as practical as is consistent with geometrical accuracy. Proposition 1.- Problem. On a given finite right line (AB) to construct an equilateral triangle. Solution. With one extremity (A) of the given line as centre, and the given line as radius, describe a circle (B C D F): with the other extremity (B) as centre, and the same radius, describe a circle (A C E F). From a point (C) in which these circles intersect, draw right lines (CA, C B) to the centres. These right lines with the giren line include an equilateral triangle (ACB). Demonstration. One of the drawn lines (CA), and the given line (A B), are radii of the same circle (BCDF): they are therefore equal, by Def. 32. The other drawn line (CB), and the given line (A B), are radii of the other circle (A CEF), and are equal. Thus each of the drawn lines is equal to the given line: consequently, by the first axiom, all the right lines (C A, CB, A B) are equal; that is, the triangle (A B C) is equilateral. There is manifestly another point of intersection (F), from which, А. E F were lines drawn to the centres, they would form, with the given line, another equilateral triangle. On account of the simplicity and great utility of this proposition, we have, with Euclid, made it the first of the series. Proposition 2.—Theorem. If two triangles (A B C D E F) have two sides (A B, A C) of the one respectively equal to two (DE, DF) of the other; and the angles (AD), contained by these sides, equal; then the triangles are equal in every respect, and have the angles equal which are opposite to equal sides. (B=E, and C=F.) Demonstration. If one of the triangles (BAC) were laid on the other, so that the vertices of the equal angles (A D), as well as one pair of equal sides (A B, D E), might coincide, the other pair (AC, DF) would coincide, because of the equality of the B angles (A D); the extremities of each pair of equal sides, remote from the vertex, would, of course, coincide (B with E, and C with F); but these are the extremities of the bases or third sides (BC, EF): hence, by Def. 7, the bases would coincide, and are equal. (Axiom 5.) Thus the triangles would be wholly coincident, and are equal in every respect. The coincident and therefore equal angles are evidently those which are opposed to equal sides. Proposition 3.— Theorem. If two sides (A B, A C) of a triangle are equal, the angles (C B) opposite to them are equal. Demonstration.-For, if the triangle were reversed, the vertex (A) retaining its position, so that each side would replace the other, the position primarily occupied by either extremity of the base would now be occupied by the other, and either angle at the base would precisely fill the same space which previously contained the other : hence, these angles are equal. Corollary.—An equilateral triangle is equiangular. For, by this proposition, the angles opposite to every pair of equal sides are equal. Proposition 4.-Theorem. If two angles (B C) of a triangle are equal, the sides (AC, AB) opposite to them are equal. (See the last figure.) Demonstration. If the base were reversed, end for end, since the angles are equal, the sides would precisely interchange positions : they would, consequently, meet at the same point as before in the plane of the triangle; that is, the vertex (A) would retain its в position, and each side would occupy the space left by the other. These sides are therefore equal. Corollary.—An equiangular triangle is also equilateral. Euclid's first book, the former being the celebrated pons A. G. ALGEBRA.- Addition and Subtraction. The ADDITION of algebraical quantities is the adding together of two or more terms, so as to form one expression. The following is a sum in addition, although only the order of the terms is altered : Sum = ax -- by + C2 Similar terms, however, must be reduced to one term by adding their coefficients : thus, 5a - 38 Sum 9a - 105; where 5a and 4a are evidently equal to 9a; and 36 to be subtracted together with 76 to be subtracted, amount to 106 to be subtracted, or -106. The process is identically the same, even when similar terms have different signs : as is the following, 5a + 36 4a — 76 Sum 92 — 4b; where the result is different, although the terms are the same as in the preceding operation : the reason of this may be readily explained : Whatever a positive term represents, its corresponding negative represents the contrary: if, for instance, £30 signifies thirty pounds in hand, - £30 signifies thirty pounds' debt. Thus, if a man owes a debt of £30, but has £70 on hand, he is, in fact, worth £70 – £30; that is, £40. If, on the other hand, he owes his creditor £70, and has only £30 wherewith to pay, he is worth £30 — £70; that is, — £40: not only is he penniless, but he owes £40 to boot : he is, in fact, £40 minus. A minus quantity is, therefore, (philosophically speaking,) a quantity less than nothing. ex + 3 If these principles be understood, the reader will see that the answer to the following sum is correct : 4a' - 3bc - ex + y Ta! + 5bc # +3+Y - 2. Here, as was required, similar terms have been reduced to one term by the addition of the coefficients. It will be seen that the — ex of the first line, together with the - ex of the third, cancel the lex of the second line, and the result is nothing. As, a man who owes his creditors the very same amount as they owe him, gives nothing, and gets nothing: the debts are cancelled. SUBTRACTION, or the taking away of one quantity from another, is performed, by simply changing the sign of the quantity to be subtracted, and then proceeding as in addition. Thus, to subtract 56 from 96, 96 56 Difference 9b- 5 = 46. 96 - 56 Difference :96-(-56)=96+56=146. The reason of this will be obvious, on the consideration that, if the subtraction of one quantity from another produces a small result, while the subtraction of nothing produces a still larger result, the subtraction of less than nothing must lead to one even larger still. M. L. R. ASTRONOMICAL PHENOMENA. MARCH, 1855. By A. GRAHAM, Esq., Markree Observatory, Collooney. MERCURY will be in inferior conjunction with the Sun about half an hour before noon of the 6th. As he will then be near that part of his orbit most remote from the ecliptic, on the northern side, he will, relatively to the Earth, be several degrees northward of the Sun's disc, in passing from the eastern to the western side. He will be nearest to us on the 9th. His apparent motion among the fixed stars will be westward till the 18th, at 9h. in the afternoon, when he will be stationary. After this his motion is direct. On the 24th, a little after midnight, he will be in the plane of the ecliptic, going southward. At the close of the month he will have receded so far westward, as to rise an hour and a half before the Sun. Venus is increasing in brilliancy, and apparently receding from the Sun. Her form is gibbous. She will be near the Moon on the evening of the 19th. Mars is close to the Sun, approaching him on the eastern side. JUPITER is also close, receding on the western side. The latter, towards the end of the month, will be visible to the naked eye before sunrise. SATURN is slightly diminished in magnitude, in consequence of his increased distance. The ring retains nearly the same proportions, the outer axes being 40' and 181. The diameter of the sphere is only 161 less than the minor axis. The Spring Quarter commences on the 21st, at 4h. Sm. in the morning. RISING AND SETTING OF THE SUN, FOR THE PARALLELS OF THE BRITISH ISLANDS. LAT. 50° 52° 54° 56° 58° Rises. Sets. Rises. Sets. Rises. Sets. Rises. Sets. Rises. Sets. h. m. h. m. h. m. h.m. h. m. h. m. h. m. h. m. h. m. h. m. March 16 45 5 42 6 47 5 396 50 5 36 6 53 5 336 56 5 30 116 24 5 58 6 25 5 57 6 26 5 56 6 27 5 546 29 5 53 21 6 2 6 146 2 6 1416 1 6 15 6 1 6 15 6 1 6 15 31 5 40 6 30 5 39 6 31 5 37 6 33 5 35 6 35 5 32 6 38 SUN AND PLANETS AT GREENWICH. Sun. MERCURY. VENUS. MARS. JUPITER. SATURN. URANUS. Sets. Sets. h. m. h. m. Rises. Sets. Rises. 7 52 7 20a. 6 24a. 6 29 8 24 6 34 Rises. Sets. Sets. h. m. h. i. 1 54m. 11 33a. 1 17 10 56 041 10 19 06 9 43 PHASES OF THE MOON. Full 3d day, 10h. 8m. aftern. Last Quarter 11th day, lh. 59m. aftern. New 18th day, 4h. 45m. morn. First Quarter . 25th day, 11h. 25m. morn. H. T. & J. Roche, Printers, 25, Hoxton-square, London. |