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(1)-()' = 1, equation to the hyperbola.

course,

D

K'

a nodo (from Lat. nodus, a knot) or an oval (from Lat. ovum, If in the latter equation, 6 becomes equal to a, the equatios an egg). For the sake of some mathematical readers we is changed into that belonging to the circle, viz., may just add the equation belonging to the conchoid, viz.

w* +y'=a'. y (y) (a+y)',

These three equations may also be exhibited in the following where the double sign indicates that plus is to be taken when curious forms, by which they are most likely to be more easily the properties of the superior conchoid are investigated ; and remembered, viz., that minus is to be taken when those of the inferior conchoid are under consideration. This equation shows that the conchoid is a curve of the fourth order, because, when expanded, it

(1. becomes an equation of the fourth degree. Of the figures here shown, the first was drawn exactly according to the method above described, but the second and third figures were copied, (2) merely to give an idea of their form, and their perfect accuracy is doubtful. We would therefore advise our students to draw these figures for themselves ; the process will form an interest

(3.) ing and amusing exercise.

The common Hyperbola, one of the conic sections, described The equation to the conic sections, generally, is of the fol. in the Lessons in Drawing, fig. 36, p. 226, vol. II, is another lowing form :curve which has asymptotés, as above defined. In order to

y=mx+nx; explain the nature of these asymptotés in the easiest manner, suppose that, in fig. 4, between the two axes ax and ay of the and this equation includes all the curves known by this name,

viz., the parabola, the ellipse, and the hyperbola; but we

must stop here till we take up the subject in our mathematical Fig. 4.

In reference to the passage above quoted, which gave rise to our preceding remarks, we may join in the language of Zophar the Naamathite, and say, “ Canst thou by searching find out God? Canst thou find out the Almighty unto perfection! High as heaven, what canst thou do? Deeper than hell, what canst thou know?” And to this we may add the words of Elihu the Buzite: "Touching the Almighty, we cannot find him out; he is excellent in power and in judgment.” Nevertheless we are permitted to illustrate spiritual and eternal things by material and visible objects. Hence, in Scrip

ture, we find that God is spoken of as having eyes and hands, F

which are with him, and to us, the emblems of knowledge and power. With this example before us, we may reverently compare the existence of the Eternal, which is from everlasting to everlasting," to a straight line which had no beginning, and which has no end; and we may, in the same spirit, liken the infinite and unbending rectitude, truth and justice of the great Creator, to the directrix or axis of a curve which extends both ways to infinity, without ever deviating to one side or to the the other; for with him“ one day is as a thousand years, and a thousand years as one day;" and with him, there is'" no variableness, neither shadow of turning."

How different from all this is the condition of man! Truly

may he be compared to the curve line, of which the directior hyperbola, two straight lines a r' and x k' are drawn through is continually changing; the curve line, which depends for its the centre a, in such a manner that an and a k', the hypotenuses direction at every point, upon its relation to the great and of the two right-angled triangles A B H and Ack', are at dis- invariable axis; the curve line which, even in a state of com. tances from the centre each equal respectively to the distances parative approach to that axis, can only become parallel to it AP and ar' from that point to either of the foci F and 7 of the at an infinite distance. Hence, it is that even when we curve; then, these two straight lines, indefinitely produced, borrow a simile from the curve and its asymptoté, the metaare the asymptotes of the hyperbola, and possess the property phorical comparison between God and man falls infinitely of continually approaching the four branches of the curve in short of the real state of the case ; for, although it may be as many different directions, without ever meeting them, though admitted that, during the endless agee of eternity, the purified both were continued to infinity. In like manner, if d e be and saved soul of man shall be continually drawing nearer the transverse axis of another hyperbola, having the two foci f and nearer unto God, through the contemplation of his visible and f', the same asymptotés a u' and x'k' are the asymptotes and ineffable glory, yet there will be such an inconceivably of this hyperbola. In speaking thus of these hyperbolas, we great distance between the Creator and the created, that the have considered the two opposite curves passing through the comparison dwindles down to that of continually approximatvertices B and c, as one hyperbola ; and the two opposite ing parallels, as infinite in their mutual distance as they are curves, which pass through the vertices D and E, as another endless in their mutual approach, and everlasting in their hyperbola; but, the former are frequently called opposite asymptotic relation to one another. Yet the apostle John, hyperbolas, and the latter conjugate hyperbolas.

speaking to believers, says, “we shall be like him, for we shali

see him as he is ;" true, we shall be like him in kind, but not in The equation belonging to the hyperbola, is of the following degree. His greatness, his goodness and his holiness shall fill form, viz.,

the universe ; ours, although similar in kind, shall be an 6?m*--aaya-aol2 ;

infinitesimal only in magnitude, and shall fill only that sphere.

to which he shall appoint us. Our respective spheres may a form which is very similar to the equation belonging to the enlarge as eternal ages roll on, but the mighty sphere of the ellipse, another conic section, viz.,

Eternal is as unlimited as his duration, and as comprehensive

as that infinite snace which he eyer continues to fiú with his o° °o°y=a*.

lown glory.

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LESSONS IN BOOK KEEPING.-No. IX.

(Continued from page 146).

In the following Day-Book, the entries of the Purchases and chases, in the Invoice Book inward; and those relating to Sales, Sales of Cotton, detailed in the Memoranda of Transactions, are in the Invoice Book outward; the former consisting of the here entered in the proper DR. and CR. form, and in business they actual invoices sent in with the goods, which are usually posted would constitute the original record of these transactions. The in a Blue Paper Book; the latter consisting of exact copies original documents relating to these transactions would, of of the actual invoices sent out with goods. course, be found in the Invoice Book; those relating to Pur

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Spencer and Co. Dr. to Cotton Account

Sold 14 bags of Maranham Cotton, Net 4350 lbs. at 9d. per 1b.... Discount it per cent.

16th. Thompson and Co. Dr. to Cotton Account

Sola 24 bags of West India Cotton, Net 7460 lbs. at 8 d. per lb. Discount 2} per cent.

22nd. Althorpe and Co. Dr. to Cotton Account Sold 12 bags of West India Cotton, Net 4240 lbs at 8d. per lb ..

24th. Cotton Account Dr, to Baring, Smith and Co.

Bought 30 bags of Demerara Cotton, Net 9218 lbs. at 75d. per lb.

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I may

Present Tense Present. wollen, to be wollend, willing. willing

he may

PLUR. SING.

be willing.

you will.

PLUR. SING.

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willing

be willing.

Perfect Tense. Perfect. gewollt haben, to gewollt, have willing.

willed.

(3 er Hat

PLUR. SING.

gewollt,

been willing

geroollt,

2 ihr habet

you have

Present Tense.

Present Tense. id will, I will. ich wolle, 2 du willst, thou wilt. du wolleft,

thou mayst er will, he will. er wolle, 1 wir wollen, we will.

wir wollen, we may 2 ihr wollet,

ihr wollet, you may 3 fie wollen, they will.

fie wollen,

they may Imperfect Tense.

Imperfect Tense. ich wollte, I was

ich wollte, I might 2 du wolltest, thou wast bu wolltest, thou mightst er wollte, he was

er wollte, he might wir wollten we were wir wollten, we might 2 ihr wolltet, you were

ihr wolltet, you might fle wollten, they were fie wollten, they might Perfect Tense.

Perfect Tense.
id habe I have ich habe I may have
2 bu haft

thou hast du habest been willing,
he has er habe

&c. 1 'wir haben

we have

wir haben

ihr habet fie Baben

they have fte haben Pluperfect Tense.

Pluperfect Tense. ich hatte I had

ich hatte

I might have 2 du hattest thou hadst

bu hättest

been willing, 3 er hatte he had

er hatte 1 wir hatten

wir hatten 2) ihr hattet

ihr hattet 8 Pie hatten

fie hatten First Future Tense.

First Future Tense. First Future. ich werde I shall id werde (if) I shall be ich würde 2) du wirft

thou wilt du werdeft willing, &c. du würdest er wird he will er werbe

er würde 1 wir werden

we shall
Ewir werden

wir würden 2 ihr werdet

ihr werdet

ihr würdet 31 fte werden they will fte werden

fie würden Second Future Tense. Second Future Tense. Second Future. i ich werde I shall have id werbe (if) I shall ich würde 2 bu wirft

been wil. bu werbeft have been du würdest 3 er wird

ling, &c. er werde willing, &c. er würde 1 wir werben wir werden

wir würden 2 ihr werdet ihr werdet

ihr würdet 3 fie werden fie werben

fie würben

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(14) Remarks on wollen.

3d mag das nicht.

I do not like that. Wollen implies future purpose: thus, ich will gehen, I will (to) go, Ich möchte gern wissen, wieviel Uhr I should like to know what i. e. my purpose is to go. The expression of mere futurity would

es ift.

o'clock it is. be, ich werde gehen. Kindred to this is another signification of Ich möchte wohl etwas davon haben. I should like to have some of wollen: as, e will dich gesehen haben, he wills to have seen you,

it. that is, he will have it or affirms, that he saw you.

Es mog sein.

It may be.
Id mötte lieber.

I had rather; I would rather
EXAMPLES,

Möge er lange leben!

Long may he live!
Ich muß ex thun.

I must do it. further illustrating the uses of the preceding verbs.

Er müßte fich seines Betragen så. He should be ashamed of his 3d barf ef thun. I am allowed to do it.

conduct. Es dürfte vielleicht wahr fein. It might perhaps be true. Mußte es nicht so kommen? Should it not so have happened? Es dürfte wohl geschehen.

It might easily happen. Wenn ich sterben müßte, so würde ich If I should die, I would not do Du darfst e8 nur forbern. You need only ask for it.

es nicht thun.

it. &r fann weder lefen nocy Foreiben. He can neither read nor write. Ich wollte gerne gehen.

I would willingly (i. e. would 3d tann mid irren.

I may be mistaken. 30 tonnte ihn nicht serstehen. I could not understand him. 3ch will zu Fuße gehen.

I will go on foot. Sonnen Sie heute zu mir tommen ? Can you come to me to-day? 3e wollte, daß wir gehen sollten. I was for our going.

men.

like to) go.

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