With a horrible glee, I laugh to see

Two armies in battle-array;

I mount my pale steed, to the charge I lead,
Each other to mangle and slay:

At the cannon's roar, and the streaming gore,
At each plunge of the flashing steel,

At the groans of the dying, borne down by the flying,
A hideous pleasure I feel.

With my banner unfurl'd, wide over the world,
O! a warrior brave am I;

The stout heart quails when the life-blood fails,
And the bloom from the cheek doth fly:
The mournful swell of the funeral-bell
For me is music meet;

And the widow's sigh, and the orphan's cry,
With grim delight I greet.

O! a murderous band I have at command,
My way on earth to prepare;

In fierce array, they wait for their prey,
My terrible will to declare:

In vain the heart bleeds, and affection pleads,
Unsever'd may be its tie,

Immovably fix'd is the awful decree,

That all mankind must die.

The youth may set out on the journey of life,
With all that is hopeful and gay;

While his brow is unfurrow'd by time and care,

And flowers are strewn in his way:

But ere the fire of his soul expire,

Or his beaming hopes are fled,

From his bright day-dreams I snatch him away,
And number him with the dead.

The strong man I seize in the midst of his toil,

And lay on the bed of woe;

His friends stand around, and their grief knows no bound, As he tosses him to and fro:

The scenes of the past grow dim on his sight,

And eternity rises to view,

His features grow white, and his soul takes her flight,

As he breathes forth a last adieu.

The weak when they feel my iron grasp,
May attempt to move me with tears;
The miser old may offer his gold;

'Tis vain-for I mock at their fears:
The humbled earth, for six thousand years,
With pitiless rule I have sway'd;
For never was banner so wide as the pall,
Nor sceptre so fear'd as the spade.

When a conqueror falls, his laurels fade;
And, though fresh be the field of his fame,
As time rolls on, his glory is gone,

And forgotten the hero's name:

But each passing day, as it's borne away,
More and more emblazons my fame,

And each mortal cut down, as a gem in my crown,
With kingly power I claim.

O! far and wide, the monuments rise

Of the terrible deeds I have done;

The sepulchral stones o'er the mouldering bones
Are trophies of victories won :

On, on I shall go to the end of time,

To blast, and wither, and slay;

For mine are conquests that never fatigue,
And pleasures that never decay.

Witney, Oxon.

W. S. HORTON,

LESSONS.

GEOMETRY.-Axioms. 1. Things which are equal to the same thing are equal to one another.

2. If equals be added to equals, the sums will be equal.

3. If equals be taken from equals, the remainders will be equal. This might be deduced from the former, but it is not less simple. 4. The whole is greater than its part.

This is implied in the terms whole and part.

5. Magnitudes which coincide with one another, or exactly fill the same space, are equal.

6. Two intersecting straight lines cannot both be parallel to the same straight line.

The propriety of placing this among the axioms has long been questioned, as it is thought by some not to be self-evident. It

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 (A B) 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 (BCDF) with the other extremity (B) as centre, and the same radius, describe a circle (ACEF). From a point (C) in which these circles intersect, draw right lines (C A, C B) to the

E

A

centres. These right lines with the given line include an equilateral triangle (A C B).

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 (C B), and the given line (AB), are radii of the other circle (A C E F), 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, C B, A B) are equal; that is, the triangle (A B C) is equilateral.

There is manifestly another point of intersection (F), from which,

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 BC, DEF) have two sides (A B, A C) of the one respectively equal to two (D E, D F) of the other; and the angles (A D), 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 CF.)

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, DE), might coincide, the other pair (A C, DF) would coincide, because of the equality of the B.

F

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 (B C, E F): 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 occu

pied by either extremity of the base would now be B

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 (A C, A B) 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

position, and each side would occupy the space left by the other. These sides are therefore equal.

Corollary. An equiangular triangle is also equilateral.

These two propositions (3 and 4) are the fifth and sixth of Euclid's first book, the former being the celebrated pons asinorum. The demonstrations here given are much easier than those in Euclid's "Elements," and not less convincing.

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 :

Similar terms, however, must be reduced to one term by adding their coefficients: thus,-

5a-3b
4a-7b

9a-10b;

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,

ба + 36
4a-7b

9a-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.