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Trapezium. A trapezium is a figure which has four unequal sides, and oblique angles, the area of which is found by the following

RULE.

1. Draw a diagonal line from one oblique angle to its opposite.

2. Drop a perpendicular from each of the other angles to the diagonal line, and take the length of all the lines thus formed.

3. Multiply the sum of the two perpendiculars into the length of the diagonal line, and half the product will be the answer. Thus:

The figure, A, B, C, D, represents the trapezium. The diagonal line, D, B, is 80 feet. The perpendicular E, C is 28ft. and the perpendicular, A, L, is 20 ft.; what is the area? 28+20=48X80=3840÷2 1920. Ans.

D

A

B

REMARKS, &c.-LESSON 8.

Extracts exhibiting the correct use of allegory. Allegory. This figure may be usefully employed both in serious and instructive subjects. In former times, it was the favourite method of imparting moral and useful knowledge:-of such is the nature of fable and parable.—

"Thou hast brought a vine out of Egypt; thou hast cast out the heathen and planted it; thou preparedst room before it,and didst cause it to take deep root, and it filled the land.

The hills were covered with the shadow of it, and the bows thereof were like the goodly cedar.

She sent her boughs into the sea, and her branches unto the river. Why hast thou then broken down her hedges, so that all they that pass by the way do pluck her? The boar out of the wood, doth waste it, and the wild boar of the field doth devour it.

Return, we beseech thee, O God of hosts!-look down from heaven, and behold, and visit this vine,-and the vineyard which thy right hand hath planted, and the branch thou didst make strong for thyself."

"Did I but purpose to embark with thee

On the smooth surface of a summer sea,
While gentle zephyrs play in prosperous gales,
And fortune's favours fill the swelling sails;
But would forsake the ship and make the shore,
When the wind whistles and the tempests roar?
No! Henry,-no!

"No, 'tis slander,

Whose edge is sharper than the sword, whose tongue
Out-venoms all the worms of Nile, whose breath
Rides on the posting winds, and doth belie

All corners of the world: kings, queens, and states;
Maids, matrons;-nay, the secrets of the grave!

SPELLING.--LESSON 9.

ma-tu-ri-ty mă-tū'rē-tē no-bil-i-ty. nō-bil'-lē-té me-chan-ical mê-kăn ê-kăl non-sen-si-cal nòn sěn sẽ ki} me-dic-i-nal mē-dis'ē-nål no-vi-ci-ate

no-vish'ē-āte

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me-rid-i-an me-ride-ăn o-bliv-i-on

ō-bliv'vē-un

me-thod-i-cal

mē-thod'ē-kǎl ob-scu-ri-ty ŏb-skū're-tê

me-ton-y-my më-ton'ē-mê ob-se-qui-ous ŏb-sē kwē-us me-trop-o-lis me-trop'pō-lis ob-ser-va-ble ŏb-zĕr/va-bl mil-len-ni-um mil-len'nē-um ob-strep-er-ous ob-strep pĕr-us mi-nor-i-ty mē-nor'ē-tē oc-ca-sion-al õk-ka zhun-ăl mē-nu'shē-ǎ oc-tag-o-nal oh tăng g0-nă mi-rac-u-lous mē-rǎk'kū-lus of-fi-ci-ate of-fish'ē-äte

mi-nu-ti-a

mis-an-thro-phymis-ăn'thro-pe of-fi-cious-nessof-fish'us-nes mo-bil-i-ty mō-bil'lē-të om-nip-o-tenceom-nip'pö-tense

549

mo-nar-chi-cal mō-nàr’kē-kăl om-ni-sci-ence om-nish ́ē-ense mo-nop-o-lise mo-nop'po-lize o-pac-i-ty ō-păs ́se-tê mo-not-o-nous mô-not'o-nus op-pro-bri-ous op-prō'brē-ùs mo-not-o-ny mɔ-not'tō-në o-rac-u-lar ō-rǎk'ku-lăr mu-nic-i-pal mi-nis'ê-pal or-bic-u-lar òr-bik ́kū-lăr mu-nif-i-cence mu-niffe-sense o-rig-i-nal

ô-rije-năl

mys-te-ri-ous mis-te'rē-ús or-thog-ra-phy òr-thog'gra-fe

my-thol-o-gy mé-thōllō-je

ne-ces-si-ty nē-ses'sê-tē

ne-fa-ri-ous ne-go-ti-ate neu-tral-i-ty

nē-tā rē-us

os-ten-si-ble Ŏs-těn ́sé-bl

o-vip-a-rous ō-vip'pā-rus

pa-rab-o-la pā-rāb ́bō-lā ne-go'shë-âte pa-ren-the-sis pā-ren'the-sis nu-trál'ē-të par-he-li-on pär-he'lé-un

LESSON 10.

John Adam's reply, &c. continued.

6. If we fail to support this declaration of independence, it can be no worse for us. But, we shall not fail. The cause will raise up armies: the cause will create navies; the people, if we are true to them, will carry us gloriously through the struggle. I care not how fickle other people have been found; I know the people of these colonies; and I know that resistance to British aggression is deep and settled in their hearts, and cannot be eradicated. Every colony, indeed, has expressed its willingness to follow, if we would only take the lead.

7. Sir, the declaration will inspire the people with increased courage: instead of a long and bloody war for the restoration of privileges, for redress of grievances, for chartered emunities, held under a British king, set before them the glorious object of entire independence, and it will breath into them anew the breath of life.

8. Read this declaration at the head of the army;-every sword will leap from its scabbard, and the solem vow rise to heaven to maintain it, or perish on the bed of honour. lish this declaration from the pulpit; religion will approve it, Puband the love of religious liberty will cling round it, resolved to stand or fall with it. Send this declaration to the public halls; proclaim it there; let them hear it who heard the first roar of British cannon; let them see it, who saw their brothers, and their sons fall on the height of Bunker Hill, and in the streets of Lexington and Concord,—and the very walls will cry out in honour of its support.

9. Sir, I know the uncertainty of human affairs; but, I see, I see clearly, through this days business. You and I, indeed,

may rue it.

We may not live to see this declaration made good. We may die, die, colonists,--die, slaves;--die, it may be ignominously, and on the scaffold. Be it so. Be it so. If it be the will of heaven that my country shall require the poor offering of my life, the victim shall be ready at the appointed hour of sacrifice, come when that hour may. But while I do live, let me have a country, and that a free country.

10. But, what ever may be our fate, be assured, this declaration will stand. It may cost treasure; and it may cost blood; but it will stand, and it will richly compensate for both. -Through the thick gloom of the present, I see the brightness of the future, as the sun in the heavens. We shall make this a glorious day. When we are in our graves, our children will honour it. They will celebrate it with thanksgiving, with festivity, with bondfires, and illuminations. On its annual return, they will shed tears, copious, gushing tears, not of subjection and slavery, nor of agony and distress, but of exultation, of gratitude, and of joy.

11. Sir, before God, I believe the hour is come.-] -My judgment approves this measure, and my whole heart is in it. All that I have, all that I am, and all that I hope in this life, I am now ready to stake upon it; and I leave off as I began, that, live or die, survive or perish, I am for the declaration. It is my living sentiment, and, by the blessing of God, it shall be my dying sentiment;-Independance now, and INDEPEN

DENCE FOREVER.

LESSON 11.

7. PARALLELOPLERON.-This is nothing more than another trapezium of a different figure. It has two parallel sides. -Being the segment of a triangle, cut off by a line drawn parallel to the base. The area of which may be found by the

following

RULE. 1. Let fall a line from either of the obtuse angles, perpendicularly to the base and find its length.

2. Multiply half the sum of the two parallel sides, by the length of the perpendicular line, the product will be the area, Thus:-

In the trapezium. A, B, C, D, the side A, B, is 22 ft. the side, C, D, 12 ft. the line C, E, is 13 ft; what is the area?

Ans. 221 ft. 22+12=34÷÷2=17X13=

221 ft

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Polygram. This figure is a species of irregular polygon; it is bounded by five or more unequal sides, with as many oblique angles. The area of such figures may be found by the following

RULE. 1. Divide the figure by lines, into trapeziums and triangles as may be most convenient.

2. Find the areas of each by the appropriate foregoing rules, and the sum of all will be the answer.

Let the figure, A, B, C, D, E, F, represent a polygon, divided into the trapezium, A, B, E, F, and the two angles, B, D, E, and B, C, D, then, draw the perpendicular, B, a, B c, F, F, and D,D.-And the sum of the areas of these will produce the true area.

E

F

D

Thus:

с

a A

9. Polygons. These are figures of from 3 to 12 or more equal sides, and of as many angles. Their areas may be found by the following

RULE. 1. Produce a perpendicular from the centre of the given figure to the medial of either of the sides.

Thus:

A

2. Multiply the sum of the sides by the perpendicular, and half the product will be the area. Let the figure A, B, C, D, E, represent the polygon, each side of which is 16.4, and the perpendicular I, G, is 11.3. What is the area?

16.4X582.0X11.3=926.60 2 463.30 Ans.

E

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D G C

Any polygon may be constructed by numerical operation, by the following

RULE.

1. Divide 360 by the number corresponding with the sides of the intended polygon.

2. Then, as the quotient is to 60, so is the side of the poly-. gon required to the semidiameter of the circumscribing circle. Thus:

In a polygon of 8 equal sides, (called an octagon,) each side being 7.5 inches; what is the semidiameter of the circumscribing circle? Ans. 10 in. 360÷8=45. Then as, 45:60 :7.5: 10. semidiameter,

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