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Multiply the area of the base by 1-3 of the perpendicular altitude;--the product will be the answer. Thus:
Suppose a triangular pyramid to be 45ft. high, and each side of the base 10lt.; what is its cubic contents?
Ans. 649.5. 10X3530;-2=15-10=5
15X5=75X5=375+5=1875 And the v 1875=43.3, area of the base. Then
45:-3=15X43.3=649.5, cubic contents. Obs. 1. Suppose the figure of the pyramid to have been quadrangular, or a square base, each side 10ft. and the perpendicular altitude 45ft.; what is its cubic contents? Ans. 1500ft. 10x10=100ft. area of the base. [See square surfaces, Rule 1, page 543.] 45+3=15X100=1500 cubic ft.
Obs. 2. Suppose the given figure had been circular, and the diameter of its base 10 feet, and its perpendicular height 45 feet, the solid contents might have been found by multiplying the area of its base into 1-3 of its height. Thus:--10x10=100, square of the diameter, X.7854=78.54, area of the base, (see Obs. 1, Case 2. Circles, page 555.] Then, 45+3=15 X 78.54=1178.10, Cubic contents.
Note. The above circular figure is called a Cone, and is nearly represented by a loaf of fine sugar.
OBs. 3. The superficial content of a cone or a pyramid, maty
be determined by the following RULE. Multiply half the girth of the base by the slant bicight, and, to the product, add the area of the base; the sum will be the superficial content. Thus:
Suppose the base of the pyramid to be square, and each side iOft. and its slant height 48st.; what is the measure of its exterior surface?
Ans. 2020ft. 10+10=20+10=30+10=40ft. girth of the base, and 40 X 48=1920ft; and 10x10=100, area of the base. Then, 1920+100=2020ft.
Obs. 4. A frustum of a pyramid or a cone, is that part of either, which is left when the top is cut off by a plane, paral. l'el to the base. The solid contents of such figures may be found by the following
Rule. 1. Find the area of both extremes, and extract the square root of their product.
2. To this root add the areas of the two extremes, and mul
tiply the sum by 1-3 of the perpendicular height; the product will be the contents. Thus:
Suppose that each side of the larger base of the frustum of a square pyramid be 10ft. and the sides of the smaller base 6ft, each, and the altitude 25ft.; what is its cubic contents?
Ans. 1633. + 10x10=100, area of the greater base. 6X6=36, area of the lesser base; and 100 X 36=3600, the square root of which is 60, and
60-+-100=160+36=196.X8=1633. Obs. 5. The superficial contents of a frustum of any kind may be found by the following
Rule. Add the girth of both bases;--one half of which multiplied by the slant height, will give the curve surface; to which add the area of both bases, and the sum will be the superficial contents.
OBs. 6. The slant height of any frustum may be found by the following
RULE. As the difference between the diameter of the two bases, is to the perpendicular altitude; so is the diameter of the greater base, to the slant height.
REMARKS, &C.--LESSON 36. Fourthly. I will here observe, that when the property of use, with regard to words and phrases, appears to be divided, and either term in question is susceptible of two or more meanings, while the other has but one application, the latter has the preference: as, proposal, for a matter submitted, is better than proposition; because the latter may imply a mathematical question, or a certain position.
To purpose, a thing when it implies an intention, is better than to propose it;--which may also imply to submit or lay before.
And I mistake, is better than I am in an error', on more accounts than one.
Fifthly. In all doubtful cases, the choice must be determined by analogy. The phrase, Though he were ever so good, is more consonant to the idiom of the language, than, Though he were never so good. On each side, referring to both sides, is better than on either side. Whether he will or not, is better than will or no;—for the ellipsis of the verb requires the first form.
Sixthly. When the terms in question appear to possess perfectly equal claims, in respect to all the foregoing provisions,
then regard to simplicity and the demands of the ear, must dictate the decision. Accept my book, is more simple and pure, than to say, accept of my book. And, address him a line, is better than, address to him a line. Delicacy pleases the ear more than delicateness;—and authenticity, more than, authenficateness. Who would not exchange vindicative for vindictive?
Seventhly. Words and phrases which are harsh and void of harmony, -as well as those that are low, cantish, and inelegant, should always be rejected, though they may have the authority of use.
Unsuccessfulness, unharmoniousness, peremptobleness, holily, godlily, fc. are harsh and unpleasant terms, and readily admit of substitutes much more inviting.
I had as lief, or lives, and, I had rather, are faulty connexions;—the helping verb had, is employed for would, the legitimate conjugation. Besides, lief and lives are too common to express a preference.
Note. The foregoing remarks, fic. are such hints as I have thought proper to submit to the consideration of the learner, with a view of rendering him some assistance in the correction of his own productions. If, however, he would become an able, ready writer,--a judge of general composition,-and a just and confident critic, he must apply himself to the study of the great laws of criticism, in more extensive and systematic publications.
SPELLING.LESSON 1. cru-ci-fix-ion kro-se-fiks'yún em-i-gra-tion ěn-e-grā’shin cur-ve-lin-ear kūr-vē-lin'yăr em-u-la-tion em-x-la shăn dec-i-ma-tion děs-sē-ma'shũn en-er-get-ic en-er-gět'ik dec-la-ma-tion děk-la-ma'shŭn en-er-va-tion ěn-ěr-vā'shữn def-i-ni-tion děf-e-nishún ep-i-dem-ic ép-e-děm'ik deg-ra-da-tion dég-grā-dā'shūn ep-i-lep-tic ép-e-lép'tik det-es-ta-tion dět-ěs-tā'shūn e-qui-noc-tial e-kwē-nok'shăl det-o-na-tion dět-ő-nā'shún er-u-di-tion ēr-ū-dish'ün det-ri-men-tal dět-rē-měn'tăl ev-a-nes-cent év-ā-něs'sẽnt di-a-cou-stics di-ā-kdû'stiks eu-ro-pe-an
yu-rô-pẽăn di-ar-rhe-a di-ăr-rē'ă ex-ha-la-tion ěks-hä-lā'shun dil-a-ta-tion dil-lā-tā'shŭn ex-hi-bi-tion ěks-he-bish'un dim-i-nu-tion dim-mē-nū'sh ex-hor-ta-tion ěks-hòr-tā'shún dip-lo-mat-ic dip-lo-măt'ik ex-pi-ra-tion ēks-pe-rā'shủn dis-po-si-tion dis-po-zish'ún ex-po-si-tion ěks-po-zishún dis-qui-si-tion dis-kwē-zish'ùn cx-su-da-tion ěks-sū-da'shặn
div-i-na-tion div-e-na'shũn ex-tir-pa-tion éks-ter-på'shún eb-ul-i-tion ěb-ul-lish'ún ex-ul-ta-tion ēks-ul-tā'shún ed-u-ca-tion ěd-yū-kā'shūn fer-men-ta-tion fer-měn-tā'shún ef-fer-ves-cenceěf-fēr-věs sěnse fluct-u-a-tion flúkt-yū-ā'shủn ef-fi-ca-cious f-fē-kā'shús fo-li-a-tion fo-lé-a'shun ef-flo-res-cenceěf-flo-rěs'sčnse fun-da-men-tal fŭn-da-měntál el-le-gi-ac ěl-le-ji'ăk gen-er-a-tion jěn-ěr-ā'shún el-e-men-ial el-ė.měn'tăl glad-i-a-tor glad-e-a'tur el-e-ya-tion ěl-ē-vă'shữn grat-u-la-tion grăt-yu-la'shun el-o-cu-tion ě-7-kū'shūn grav-i-ta-tion grăv-ētā'shữn el-on-ga-tion ěl-ong-ga'shūn hes-i-ta-tion hér-ē-tē'shún em-an-a-tion ēm-a-nā'shún hi-e-rarch-al hi-ē-ràrk'ăl em-bar-ca-tion ěm-bàr-kā'shūnhor-i-zon-tal hòr-e-zon'tăl em-blem-at-ic ěm-blēm-ăt’ik hy-dro-stat-ics hi-drõ-stăt'iks em-en-da-tioněm-měn-dā'shūn hy-me-ne-al hi-me-nē'āl
Conversations between a Father and his two sons on the subject
of Government, fc. Note. These conversations are said to have been had between a Mr. Brown, a thrifty farmer of the State of New York, and his two sons, Horace, a lad of fifteen, and Philo, who was about two years younger. The business of the farm had been closed, and the boys had commenced their winter studies in one of the District Schools. The subject was incidentally introduced by Horace while at the supper table with the family.
Father, said Horace, I this day read a few pages in "Goldsmith's Rome,” which was highly interesting; but there were some parts of it which I did not understand:--for instance, the word government; what does it mean?
The term, my son, has several applications; but when referred to communities of men, it very properly implies the form or manner in which the power, necessary for the administration of public affairs, is disposed of.
I suppose, then, said Horace, there are several ways of disposing of the power; but how are they distinguished?
They are distinguished, returned the father, by the different kinds of government which the different modes of disposing of the power necessarily produce. How many
kinds of government are there? asked Philo. There are only three legitimate kinds, replied Mr. Brown; Monarchical, Aristocratical, and Democratical; every other form is a mere modification of one or more of these.
We should be pleased to know how the monarchical form
of government is distinguished from the other forms? said Horace.
It is that form of government in which the supreme power is vested, unconditionally, in the hands of one person, styled a monarch, king, or emperor, whose will is the law' of the land. If he governs in a severe and arbitrary manner, he is called a despot;-and if with cruelty and oppression, he takes the name of tyrant. Should the powers of the king be limited by laws, then the government is a limited monarchy, and the laws which limit him are called a constitution. If others, such as a council, or an assembly, are associated with him in the exercise of power, it is termed a mixed government;such are the institutions of Great Britain and France.
I hope, sir, said Philo, you will now explain to us the nature of
an aristocratical government. I will, may son. An aristocracy, which is sometimes called an oligarchy, (olʻle-gratsh-ē) is that form in which the supreme power is lodged in the hands of a few of the nobility, who exercise it conjointly.
The third form, you observed, said Horace, is a democracy; how is that distinguished?
In a democracy, replied the father, the supreme power is confided to the whole body of the people. Should the people delegate that power to officers appointed by themselves for limited periods, then the form of government becomes a democratic republic.
I beg to ask, said Horace, where the supreme power comes from, and what constitutes it?
It comes from the people, said the father; they possess it from the hands of their Creator; and the portion justly exercised by government, is formed from the small parts which every man tacitly yields for the benefit of the whole, in order to constitute a sum total for government.
How much, asked Horace, does every man give up in order to make the sum possessed by government?
Just so much, my son, as will secure to him and his associates, the enjoyment of life, liberty, property, and the pursuit of happiness, in the most effectual manner. All that is claim. ed or exercised by government beyond this, is manifest usutpation.
What form of government do we live under? asked Philo.
Ours, said Mr. Brown, is a democratic republie, of the confederate order:--for, all political power with us, is in the