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16. The sector of a circle, is the space contained between two radii, and an arc less than a semicircle. As, E, C, B, in the last figure.
17. The sine of an arc, is a line drawn from one end of the arc, perpendicularly to the radius, or the diameter produced through the other end;-or, it is half the chord of double the arc. Thus:-H, Z, in the subjoined figure, is the sine to the arc H, B.. 18. The sines on the same diameter
K increase in length, until they reach the
DI centre, and then become the radius;--D, C, or the semi-diameter of any circle,
G is the greatest possible sine; and it is al А ways equal to 90 degrees.
E G 19. The versed sine of an arc, is that part of the diameter, which lies between the sine and the circumference. Thus:-Z, B, is the versed sine of the arc, H, B.
20. The tangent of an arc, is a right line, touching the circumference at one end of the arc, and rising perpendicularly to the diameter, until it terminates in the line drawn from the centre through the other end of the arc.
Thus:--K, B, is a tangent to the arc, H. B.
Note. 2. The tangent of an arc of 45 degrees, is equal in length to the radius of the circle, of which the arc is a part.
21. The secant of an arc, is a right line drawn from the centre, through one end of the arc, until it unites with the tangent. Thus:--K, I, C, is the secant to the arc, H, B.
22. The complement of an arc, is what the arc wants of 90 degrees, or a quadrant. Thus:--H, D, is the complement 01 che arc,
H, B. 23. The supplement of an arc, is what the arc wants of 180 degrees, or a semicircle. Thus:-A, D, H, is the supplement to the arc, H, B.
24. The sine, tangent, and secant, of the complement of any arc, are regarded as the co-sine, co-tangent, and co-secant of the arc. Thus:-F, H, is the sine, D, 1, the tangent, and C, I, the secant of the arc, H. D; and they are the co-sine, co-tangent, and co-secant of the arc, H, B.
25. The measure of an angle, is the arc of a circle, contained between the two lines which form the angle; the angular point being the centre. Thus:--the angle, H, C, B, is
ineasured by the arc, H, B, and the angle contains as many degrees as the arc is found to be parts of 360, the whole circle.
REMARKS, &c.-LESSON 8.
Faulty Composition. 4. If men were always prosperous they would be likewise always light headed and if they were always unfortunate they would be cast down and desponding an equable compound therefore of lights and shades hopes and fears joys and lamenting are providently blended with his being so as to give him a proper impetus in the pursuit of worldly concerns and to bring him back occasionally to hearken to the admonitions of conscience and the concerns of a coming state our keeping is in the hands of him who does all things well the employment of present time is a theme which all should turn their attention to but more specially the young what are we now doing what is the range and object of our present pursuits and the scope of our designs and intentions are our cares and inquiries and their ultimate tendency and bearing all of a complexion that will warrant thc means which we use and employ to bring them about are they likely to produce effects that will survive the maw of time and bear fruit for futurity
if we rise early and retire late and eat not the bread of idleness but do with diligence what ever our hands findeth to do yet have no respect to the good or bad results which our performances tend to we act without reflection or thought or wisdom and possibly might better be idle hence we should not only do according to divine precept what ever our hands find to do but we should be careful and do especially what will tend to the promotion of our own or our neighbours good otherwise our doings will be a dead weight at our hands through the journey of life and tarnish the purity of our celestial robe as we equip for another world nor are we allowed to waste our time in arriving at conclusions and adjusting the proper tendency of our actions we must use the present moment without abusing it each moment tells one and the aggregate of moments sum up the whole of life which every days experience shows has no returning tide embark therefore on the flood as it reaches you or you are left in an ebb that will never float your burden
SPELLING.-LESSON 9. in-ter-dict in'těr-dikt, prohibit-pre-cip-i-tate prē-sip'pě-tāte, ed.
headlong. in-ter-dict in-ter-dikt', to forbid. pre-cip-i-tate pré-sip'pē-tát, in-ti-mate în'tē-māte, to hint. medicine.
[firm. in-ti-mate in'te-măt, familiar. pred-i-cate prěd'de-kāte, to afi-ron-y i'urn-ē, made of iron. pred-i-cate prěd'dē-kāte, what i-ron-y i-rūn-ē', rhetoricial fig. is affirmed.
prel-ude prělūde, introductory low-er lö’úr, to bring low. pre-lude pre-lūde', an introlow-er lòû'úr, to frown.
duction. min-ute min'nit, sixty seconds. prem-is-es prěm'is-ěz, lands, mi-nute mē-nūte, small.
&c. inis-con-duct mis-kon'dūkt, ill pre-mis-es pré-miz'zēz, exmanagement.
plains first. mis-con-duct mis-kön-dūkť, to pres-age prés'sāje,a prognostic manage illy.
pre-sage pré-sāje', to forebode mod-er-ate mod'děr-ặt, tempe-pres-ent prěz-zēnt, not absent. rate.
pre-sent prē-zēnt', to offer. mod-er-ate mòd'děr-āte, to reg-prod-uce prod'dūse, gain, aulate.
mount. ob-ject ob'jekt, a thing sought.pro-duce pro-dūse', to exhibit. ob-ject ob-jékt', to oppose. proj-ect proj'ěkt, a scheme. out-work òût'wūrk, fortification pro-ject pro-jekť, to throw out. out-work òût-wūrk', to work prol-ate prolate, flat.
pro-late pro-lāte',to pronounce 0-ver-flow o'vur-flo, inundation pros-trate pros'trāte, lying at 0-ver-How Ö-vŭr-fo', to deluge length. 0-ver-throw Ö-vūr-t'hrõ, des- pros-trate pros'trāte, to throw truction.
down. o-ver-throw ő-văr-t'hro', to des-prot-est protěst, declared obtroy.
jection. pen-dant pěn'děnt, an ear ring, pro-test pro-test', to object. pend-ant pěn'ănt, a ship's flag.prov-ost prov'vost, chief of a per-fume për füme, sweet o body. dour.
pro-vost pro-vo', an officer. per-fume pěr-fūme', to scent. ra-ven ra'v'n, a large black per-mit pěr'mit, a passport.
[dily. per-mit për-mit', to allow. ray-en răv'v'n, to devour greeprec-e-dent prěs'sē-dent, ex- reb-el rěb'ěl, one who rebels. ample.
re-bel re-běl', to revolt. pre-ce-dent pré'se-dent, going rec-ord rěk'òrd, a register. before.
re-cord re-kòrd', to register.
READING EXERCISES, &C.--LESSON 10.
The Pilgrim's Song:-UPHAM.
The blue waves curl beneath the gale;
We leave behind our native shore,
Our homes and all we lov'd before.
The storm spread out its wings of wo,
And yet while life with us shall last,
From England's shore we'll speed us fast.
Than dwell where mind cannot be free;
Ye blasts of heaven, onward sweep,
And bear us o'er the troubl’d deep!
Another land, and other skies!
For here at last our feet shall rest,
Our minds be free, our homes be blest.
Their green boughs o’er the mountain's head;
So long shall this fair country be
The proud retreat of liberty.
A pēa'-ăn loud of sacred praise;
For fairer lands have met our view :
Old England's shores, a long adieu.
Definitions. 26. The sine, tangent, and secant of an arc, are also the sine, tangent, and secant of the angle whose measure the arc is.
Note. 1. An angle is said to be great or small in proportion to the extent of the opening of the lines which form it; or in proportion to the number of degrees embraced in the arc formed by the interception of those lines. Hence, it follows, that the magnitude of an angle does not depend upon the length of the including lines; for all arcs described on the same point and intercepted by the same right lines, contain the same number of degrees whether the radius be long or short. 27. Parallel lines are such as are equal. A.
B Ty distant from each other. As, AB, C D.
28. A triangle is a figure bounded by three lines. As, A, C, B.
Note. 2. When all the lines are equal, the figure is called an oquilateral triangle.
If the figure has one right angle, it is then called a right angle triangle.
If it has one obtuse angle, it is then called an obtuse angle triangle: but an acute angle triangle, has all its angles acute.
Obtuse and acute angles are generally styled oblique angles, one side of which is termed the base, and the others, the legs.
B 29. In a right angle triangle, as A, C, B, the longest side is called the hypotenuse, and the others, the legs, or base and perpendicular.
А. Note. 3. The three angles of every triangle being added, amount to 180 degrees; therefore, the oblique angles of a right angle triangle, amount to 90 degrees, and the right angle is always 90, 30. The perpendicular height of a
D triangle is a line dropped from one of its angles to its opposite side. Thus:the line D, C, in the angle A, B, D, is
B the perpendicular height thereof. A
31. A square figure, bounded by four equal sides, contains four right angles. As, A, B, C, D, and a line from one angle to its opposite, is called a diagonal. As, A, C.