Imagens da página

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.


'Twas a strange night;—the moon's beclouded beams

Threw a chill lustre o'er the deadly scene;
And, bursting forth anon in fitful gleams,

Disclosed in many a form what life had been :
For many a corpse the gory ground bestrew'd,
And many a brow death's clammy sweat bedew'd.

Full many a guardian spirit throng'd the air,

And bent in silent pity o'er its dead ;
But there was one, a wretched mourner, there,

Whose heart with grief's relentless torture bled :
'Twas a lone widow, sorrowing for her son,
Her only hope, her own beloved one.

Her step was hurried, and her look forlorn,

Her dark eye fill'd with an unnatural light,
Her raven hair in reckless tresses torn ;

Her wails rang plaintive through the howling night;
Swifter and swifter o'er the field she flew,
And frantic as the winds that o'er her blew.

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
She stops : the mother gazes on her son ;
But, 0! life's fleeting course is well nigh run!
He lies on the hard ground: the moon's cold beam

Plays on his marble features ; one by one
The red drops gush from the deep wound, and stream

About his head : the fatal work begun,
Slower and slower heaves his gentle breath,
Swister and swifter goes the work of death.
His is a tall and strangely haughty form ;

His brow is noble, and his bearing high
Even in Death's cold hand : yet many a storm

Has beat upon him, and life's dreary sky
Shrouded in clouds the face of hope's fair sun,
Blighting the bud of glory scarce begun.
See, she bends over him, her loving eyes

Fillid with a terrible glare, and her pale cheek
Furrow'd with grief's deep pang: “Awake,” she cries,

“And bless thy mother; wake, sweet boy, and speak;
Look-look on me, if life its presence keeps,-
Look-look on me! for, 0! thy mother weeps.”
Watch now the opening eyelid, the short quiver

About his parched lips, and the slight flush
On his damp brow; while life's returning shiver

Spreads o'er the leaden cheek a transient blush,
As, with a voice Death scarce avails to smother,
He utters in a gentle whisper,—“Mother!"
“My son, my son, that happy sound repeat,

And warm me once more with those loving eyes;
Look on me, ere my heart has ceased to beat;

Say it again, before thy mother dies !”
But, no! his heart was ever still, and run
Was the short course of the poor widow's son.
The winds grew silent:-and a piercing scream

Shook the cold earth, and rent the listening air ;
The moon in terror veil'd her silver beam ;

For there she lay,--the beautiful, the fair :
The transient blush from her boy's brow had fled,
And the young widow slept among the dead.

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-
posed on the other, the two lines including the former could,
without altering their mutual inclination, be made to coincide
with the two lines including the latter. From this it appears
that, in order to the equality of two angles, it is not necessary
that the lines forming them should be of the
same length. Now let two equal angles be
placed so that the vertices and two lines shall
be coincident, and the other two in the same
plane with the coincident lines and on opposite
sides. The two lines most divergent will now
form an angle double of one of the former angles. On the same
principle we can, in imagination, form an angle triple, quad-
ruple, &c., of any given one, an angle equivalent to the sum or

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.



Def. 12. An acute angle is less than a right angle.
An angle is designated either by a letter placed at

the vertex, or by three letters, of which the
middle one is at the vertex, and the two others
are anywhere along the sides. Thus the angle
in the margiu may be referred to either as the
angle A, or as the angle BAC. The latter mode is indispensa-
ble when several angles have the same vertex. Thus, in the
diagram, three angles have the same vertex A:
that formed by the lines AB, AC, and which
is designated BAC; that formed by AC, AD,
designated CAD; and, lastly, the one formed
by AB, AD, which is the sum of the two

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.

[ocr errors]


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.

« AnteriorContinuar »