The first volume of Studies from History, by the Rev. W. H. Rule, (Mason,) presents Richard I. of England, and Mohammed II. of Turkey, with a considerable mass of information scrupulously drawn from the original authorities. In the series of “Studies,” of which the first two are now published, it is proposed to exhibit eminent personages and events with a completeness of detail that cannot be attained in compendiums of civil or ecclesiastical history, however skilfully they may be prepared, and also without the encumbrance of extraneous disquisition which is too frequently employed to supplement single biographies. A Survey of the Geography and History of the Middle Ages, through a period extending from the year 476 to 1492, by WILHELM Putz, (Varty and Owen,) is a small, but comprehensive and extremely useful, book of reference for information that readers of mediæral history absolutely need, but often find it difficult to obtain. POETRY SCENE FROM A BATTLE-FIELD. Threw a chill lustre o'er the deadly scene; Disclosed in many a form what life had been : Full many a guardian spirit throng'd the air, And bent in silent pity o'er its dead ; Whose heart with grief's relentless torture bled : Her step was hurried, and her look forlorn, Her dark eye fill'd with an unnatural light, Her wails rang plaintive through the howling night; Yet she wept not ;-a deep untold despair Had worn her tender frame; the delicate brow, Deep furrow'd, spoke it, and a burning glare From her wild eye as she flew past: but now Plays on his marble features ; one by one About his head : the fatal work begun, His brow is noble, and his bearing high Has beat upon him, and life's dreary sky Fillid with a terrible glare, and her pale cheek “And bless thy mother; wake, sweet boy, and speak; About his parched lips, and the slight flush Spreads o'er the leaden cheek a transient blush, And warm me once more with those loving eyes; Say it again, before thy mother dies !” Shook the cold earth, and rent the listening air ; For there she lay,--the beautiful, the fair : M. L. R. LESSONS. GEOMETRY.-Definition 1. Geometry is the science which explains the properties of extension or magnitude, without regard to matter. The statement of any of its truths as a subject for argument is called a proposition. When a proposition exhibits some geometrical construction to be effected, it is a problem : when it proposes a truth or truths to be demonstrated, it is a theorem. Def. 2. Magnitudes are of four kinds : solids, surfaces, lines, and angles. Def. 3. A solid has three dimensions,-length, breadth, and depth. Def. 4. The boundaries of a solid are surfaces or superficies : hence, a surface has only two dimensions,-length and breadth. Def. 5. The extremities of a surface are lines : hence a line is length without breadth. Def. 6. The terminations of a line are points : hence, a point is position, but not magnitude. Def. 7. Lines which cannot have two points in common without being wholly coincident, are called straight or right lines. A right line has been defined, "the shortest way between two points.” This assumes that there is a shortest way, which would seem to require proof, and can be proved on principles which shall be laid down. It is from this property, that the right line terminating in two points is called the distance between those points. Def. 8. If a right line cannot meet a surface in two points without lying wholly in it, that surface is plane. Def. 9. A plane rectilineal angle is the degree of opening or divergence of two right lines which meet in a point called the vertex of the angle. To form a correct conception of this species of magnitude, imagine one line to move from coincidence with another by revolving round their common extremity. The direction of the moving line, estimated from the centre of motion, is continually changing, and varies more and more from that of the fixed line, until it is at length diametrically opposite, when one line is the continuation of the other. At this juncture the line will manifestly have made half a revolution. Proceeding still in the same direction, it will reach its initial position of coincidence with the fixed line, after having completed one revolution. And, as the angular motion may thus proceed through revolution after revolution, without end, angular magnitude may be considered capable of indefinite increase. This view is essential to the application of geometry in the higher branches of mathematics. With regard to the relative magnitudes of angles : two of these quantities are said to be equal, when, if one were laid or super- difference of two given ones, &c. Def. 10. When one right line standing on another makes the adjacent angles equal, each of these angles is called a right angle, and each of the lines is said to be perpendicular to the other. When, as in the previous illustration, the moving line has gone through a quarter revolution, it forms a right angle with the fixed line, and forms an equal angle with the continuation of that line beyond the vertex. Def. 11. An obtuse angle is greater than a right angle. A с Def. 12. An acute angle is less than a right angle. the vertex, or by three letters, of which the former angles, and is written BAD. Def. 13. Parallel right lines are in the same plane, and do not meet, however far produced, both ways. A This is the usual definition of parallel lines. It assumes that there can be right lines in the same plane, which would never meet, though indefinitely produced in both directions. This is a truth which has been rigorously demonstrated, as shall be shown; but it seems hardly consistent with the rigour of geometrical reasoning to admit it here as a concealed axiom. It is to be observed, also, that the non-occursibility of two lines in the same plane, is not, in all cases, a criterion of parallelism. For example, a right line can be drawn which would never meet either branch of an hyperbole, however far produced ; and yet, as they continually approach, the hyperbole and the right line, so related, are no where parallel. The subject of parallel lines shall be taken up at greater length hereafter. A. G. ALGEBRA.-Algebra expresses the relations of abstract quantities. Abstract quantities may be expressed in any arbitrary manner. The letters of the English alphabet are generally used : thus, we may add a and b together, or we may subtract one from the other, or multiply or divide one by the other, without giving them any fixed numerical value. Certain signs are used to express the operations of addition, subtraction, multiplication, and division. There are other signs also. These must now be explained. + (which is read plus) is the sign of addition : thus, instead of saying, a added to b, we say, a plus b, and write it, a +b; and instead of saying, 5a with 3a added, we say, 5a plus 3a, and write 5a + 3a, and their sum is 8a. – (which is read minus) is the sign of subtraction : thus, instead of saying, a with b taken from it, we say, a minus b, and write it, -6: so, if we wish to subtract 3a from 5a, we write 5a - 3a, and the result is 2a. There are two combinations of the preceding signs, very useful but not so frequently met with : † (plus or minus) shows that the quantity to which it is prefixed may be either added or subtracted; thus, 7a+ 3a is either 10a, the sum, or 4a, the differ ence, of these two quantities. F (minus or plus) shows that the quantity to which it is prefised may be either subtracted or added ; thus, 7a 7 3a is either 42 or 10a. x (which is read into) is the sign of multiplication : thus, in multiplying a by b, we write a xb; for as a and b represent quantities, they can of course be multiplied ; if a, b, and c are to be multiplied, the process is shown thus, a x bxc. This sign, however, is frequently omitted, and a b c is precisely the same as a xb xc, just as 7a is the same as 7 times a, or 7 x a. |