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gathered for her a stem from the mezereon, during the interval before Bervice. With a look of inexpressible sweetness, full of words, she laid it on my open Bible; and I kept it long after ber death, because it spoke to me of her."

CONQUEROR of winter's power,
With thy purple sceptre flower,
Earlier than the snowdrop pale,
Or the lily of the vale;
Coming ere the dawn of spring,
Tidings of her thou dost bring,
Harbinger of better things;
Sweeter is thy bloom to me
Than the summer rose's blow,
Or the autumn peach's glow :
Emblem thou of strength in sorrow,
Hoping for a gladsome morrow;
Cheerfulness in hours of gloom,
Smiling over winter's tomb,
As were nought so bleak and bare,
But some hidden joy were there.
While I gaze upon thee now,
I remember sadly, how
Thou wast once bestow'd on me
By a being fair as thee.
Thou art here, but she is gone;
Still thy grace I look upon.
Hers is lowly laid, and grief
Twines for her a funeral wreath;
Sad,-yet hopeful, as thy bloom ;
Weaves it on her silent tomb.
Faith regards thy blossom bright,
Marks the close of winter's night;
Knows the lovelier human flower,
Blighted by earth's wintry power,
Blooms in an eternal tower :
Lost awhile to mortal sight,
Still in heaven's own garden bright,
Tended by an angel's hand,
Guarded by an angel band.
'Twas a Sabbath morning, fair,
Though no summer's sun did shine,-
For there breathes in Sabbath air
Something heavenly and Divine,-
In God's holy house we were,
For it was the hour of prayer,

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And she laid thee in my book
With a calm expressive look :
Her whole action seem'd to say,
I shall go, but this may stay ;
Whispering of me, it will say,
She is in the land of bliss,

Far, far away.''

KATHAROS.

LESSONS.

GEOMETRY.—Proposition 10.- Theorem.

If one side (BC) of an angle (ABC) be produced beyond the vertex, the angle (A B D) made by the other side (A B) with the continuation (BD) is the supplement of the original angle.

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If one is a right angle (Fig. 1), the other is right, by Def. 10.

If one is oblique, that is, acute or obtuse (Fig. 2), at the angular point erect a perpendicular (BE) to the produced side; this perpendicular divides one of the angles (ABD) into two parts, one of which (EBD) is a right angle; the remaining one (E BA) and the other angle (A B C) make up another right angle (EBC): thus, the two angles in question (A B C, ABD) are together equivalent to two right angles,

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Cor. 1. The supplement of the greater of two angles (A B C) is less than the supplement of the less (D E F).

Conceive the vertex (E) and produced side (EF) of one applied to those of the other (B and BC); the continuations (EH and G BG) would coincide; and the remaining side (ED or B K) of the less would fall nearer to the produced side (BC), and therefore farther from the continuation (BG) than would the remaining side (A B) of the greater; that is, the supplement (ABG) of the greater angle is less than (KBG or DE H) the supplement of the less.

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Cor. 2. All right angles are equal.

For, if one were greater than another, the supplement of the former would be less than that of the latter (Cor. 1), therefore less than the latter (Def. 10), therefore less than the former; which contradicts Def. 10.

This corollary is usually reckoned among the axioms : ve hare preferred demonstrating it. Cor. 3. Supplements of the same or of equal angles are equal.

Proposition 11.— Theorem. If two supplemental angles (ABC, A B D) are on opposite sides of a common leg (A B) and vertex (B), their other legs (B C, B D) are in the same straight line.

For, the continuation of one of these legs (BC) beyond the vertex makes with the common leg (AB) an angle equal to that which the other leg (BD) makes with it, by Prop. 10, Cor. 3; and therefore the continuation of BC) is coincident with the other leg (BD). This proposition is the converse of the former,

Proposition 12.-Theorem. Vertically opposite angles (B AC and DAF, or CAD and FAB), made by two intersecting right lines (B D, CF), are equal.

10

D

B

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Cor. 1. The four angles made by two intersecting right lines are together equal to four right angles.

Cor. 2. All the consecutive angles made round a point by any number of lines terminating in that point are together equal to four right angles.

For, by producing any two of the lines beyond the vertex of the angles, four angles are formed which are together equal to four right angles (Cor. 1), and they include all the angles round the point.

Cor. 3. All the consecutive angles on one side of a line passing through the common vertex in the last corollary, are together equal to two right angles. This is evident from Prop. 10.

Proposition 13.-Theorem. If two equal angles (BAC, D AF, Fig. to Prop. 12) are vertically opposite, and two sides (A B, AD) are in the same straight line, the other two (A C, A F) are in the same straight line.

For, the continuation of one of the latter sides (AC) beyond the vertex, makes with one of the former (AD) an angle equal to that which the other (A F) makes with it, by Prop. 12 and hypothesis; therefore the continuation (of AC) coincides with that other (A F). This is the converse of Prop. 12,

A. G.

66

ALGEBRA.-EQUATIONS. An equation is an expression, representing the identity in value of two quantities under a difference of form,

The term "equation" cannot be applied to two quantities identical in form. 2x = 2x is not an equation ; for a thing cannot be said to be equal to itself, without, by implication, depriving it of its identity. Nor must such an expression as 2x + 3x = 5x be deemed an equation; for it expresses an equality which admits of no doubt, and must therefore be designated an identity.”

There are, therefore, two requisites for an equation; namely, identity of value, and absolute difference of form, in the two quantities which compose it.

Equations are of use in the discovery of unknown values. If—to take a very simple example—there be a number, to the double of which by adding 4 the result is 16, we may readily find what that number is. For, representing it by x, or any other convenient symbol, we have 2x + 4 = 16; therefore 2x = 12, and x=

= 6, the number required.

Similarly, if &- a=b- 3, (x being the unknown quantity,) by adding a to both sides of the equation, we have z=a +b - X; and

a+b. again by adding x, 2x =a +b, .., dividing by 2, x =

2 Equations are of various kinds; and those which we must first notice are simple equations.

• For, if equals be added to equals, the results are equal.

.

SIMPLE EQUATIONS. A simple equation contains the first power only of the unknown quantity.

Every simple equation, with respect to the unknown quantity 1, can be reduced to the general form at +b= 0. For example, 4x + 9 = 18 can be reduced to the form 4x + (- 9) = 0; and that without destroying its identity, or throwing x out of sight. And so with any other; a, in the formularic expression just given, representing the co-efficient of x, and b the known quantities of the equation.

A simple equation can have only one solution ; that is to say, the unknown quantity in a simple equation can have but one value : and this may be proved thus,

If it be possible, let there be two values of x in the general expression ax + b =0, and let these be represented by a, B, respectively. Then aa+b=0

(i)
and aß+b=0

(ii)
.., subtracting (ii) from (i), aa - =0

or, a (a-B)=0. Now, either a, or the other factor, a- - B, is equal to nothing; but it cannot be a, or the proposed equation would be no equation at all with respect to x; consequently it is a - B which equals 0, or a=B. That is to say, the two supposed values which we assigned to I are, after all, one and the same. There is, therefore, only one value of x which satisfies the general equation ax + b = 0. The following are offered to the ingenuity of the reader :

abc 1.

b 2.

1 b

=a? +6%,
b.x

ab
3.
X + 6 16 - 3:2 25

X = 8.
4 12
4. +
3,

x = 12.
3 4 6
The first, for instance, may be solved thus :-

abx abr Multiply by ab, +

abe.

b
or, bx + ax = = abc.
or, x (a + b) = abc.

abc
and, divided by a +b,

M. LR.

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