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set off the spaces z a, ab, etc., z a', a' b', etc., six on one side and six on the other: the straight line c D, which is composed of twelve of these equal parts, will represent the equator, on either side of which the gores are to be formed, and be exactly equal to the equatorial circumference of the pasteboard sphere. Through c, z, and D, draw the straight lines EF, XY, GH, at right angles to A B. From c, on either side of A B, along C E and c F, set off cм, C K, each equal to three times z a, or half z c; and from D, along D H, DG, set off DL, D N, each equal to half z c. Join KL, MN; then KMNL is a parallelogram of which the length CD or M N is equal to twice the breadth м K; and in the lines K L, M N, the points forming the extremities of the gores should fall, or the arcs which form either side of each gore intersect.

To draw the gores, it is first necessary to obtain a line as a radius with which the arcs containing them may be described. The radius of the ares is equal to nine and a-half times z a, or DO, or CV; and may be readily determined by bisecting c d or dd', (the tenth space numbering from D or c) in o or v. With this radius, from centres on the line z A produced indefinitely towards

or better, with a radius of 25°, which will leave room for the delineation of the arctic and antarctic circles with a margin beyond them.

The gores and circles should be drawn on thin but very tough paper, and the coast-lines, rivers, mountains, etc., should be carefully sketched in Indian ink, and the lettering inserted as far as possible. They should then be damped and pasted on the sphere, care being taken to fit the gores together with exactness, so that the arcs forming the parallels may all meet. In pasting on the gores, the meridians drawn on the surface of the sphere will be found useful as guide-lines to fix the position of the gores. When thoroughly dry the lettering must be completed, and the land, water, etc., must be coloured, a coat or two of white varnish being added as a finish. The globe thus prepared must be fixed in a brass meridian circle properly gra duated, and the whole suspended in a wooden frame, the upper part of which is a circle to represent the horizon, and what is called the wooden horizon. This is covered with paper graduated to show the succession of the months, relative time at different parts of the earth's surface, etc. etc.

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B.

A, describe ares through the points D, e', d', etc., as far as e; and through the points c, e, d, etc., as far as e', describe arcs with the same radius from centres in z B produced indefinitely towards Thus o will be the centre of the arc passing through D, and Q the centre of the arc y y' passing through c; and in the same manner v will be the centre of the arc passing through c, and P the centre of the arc xa passing though c. To draw the parallels of latitude in each gore, join the extremities as x x' or y y', and divide this straight line into nine equal parts, each containing 10°, on either side of the equator, and through the points thus found draw arcs from x, x, y, y', as centres, as shown in the diagram. To draw the meridians, divide the equatorial parallel in each gore, as b c or b'd, into three equal parts of 10° each, also trisect the third arc from the equator, and the third arc from the extremity in each half-gore, and form the meridians by drawing curved lines through the extremities of each gore, and the five intermediate points determined by the trisection of the equatorial parallel and four arcs of latitude.

Although the whole of the gores should be drawn for the purpose of determining the extremities as centres from which to describe the arcs that form the parallels of latitude, the delineation of coast-line, etc., need not be carried above lines drawn through R, S, and T, U, a little above the arctic circle. Circles to finish the top and bottom of the globe may then be drawn on a separate piece of paper, as shown at x with a radius of 23°,

We now pass on to the method of drawing a projection of any small part of the earth's surface that we may wish to dolineate on a large scale. Suppose Italy were the country of which we wished to make a map, using the conical form of projection, and that we desired to draw it on the scale of 1 inch t 60 geographical miles, or a degree of the equator, it is manifest that to describe the arcs forming the parallels of latitude from a common centre, as in the projection for the map of Europe (Vol. II., p. 356), we should require to find a point distant, approximately speaking, about 5 feet from the point through which the arc representing the parallel of 36° north latitude must be described. It would be inconvenient to do this unless we were provided with beam compasses and a flat table of great length. Let us see, then, how we can construct a projection by the aid of ordinary compasses only, and yet preserve the curved form and parallelism of the arcs representing the parallels of latitude, giving each its proper degree of curvature.

As Italy covers pretty much space for our purpose, and as in naming Italy we merely wished to call attention to the great length of the radii required for describing the arcs representing the parallels of latitude from a common centre, in delineating any of the southern countries of Europe, we will take Scotland. which lies-the Orkney Islands being included-between the parallels of 54° 30′ and 59° 30′ north latitude. First of all draw a perpendicular straight line A B (Fig. 22), and through c, a point

LESSONS IN GEOGRAPHY.

near its lower extremity A, draw D E of indefinite length, and at right angles to it. The point c represents the point through which the parallel of 54° 30' would pass if it were traced in the projection. Assume a space equal to 60 geographical miles, or one degree of the equator (in Fig. 21 we have assumed this as being equal to an inch); set off half the length of this space upwards from c, and number the point thus obtained in AB as 55, and from 55 set off four of the spaces assumed, numbering the points thus obtained, 56, 57, 58, and 59. From 59 set off to F a space equal to that set off from c to 55. Now we will take the parallels of 59° and 55° north latitude as the parallels through which the circumscribing cone would cut the sphere if we were going to work to construct a projection of the conical form as for the map of Europe, and through the points thus numbered draw short straight lines parallel to DE, and also through

Fa straight line

GH, parallel to G

DE. The straight

line GH forms the northern

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The projection must be continued as far as necessary on either side of AB by repeating the process-that is to say, from m as a centre, with radius me, an are must be struck passing through the points e and s, the point s being fixed by drawing an arc from o as centre, with radius oe or ab. Similarly from o as centre, with same radius -namely, a radius equivalent to a diagonal of any of the three quadrilateral figures moef, kg hl, or fe pn-draw an arc passing through f and q, this last point q being fixed in position by intersecting the arc last drawn by another described from m

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limit of our map. It is now necessary to construct a diagonal scale on the space assumed as equal to one degree of the equator, as in Fig. 21, to enable us to determine proportionately the length of a degree of longitude under the parallels 59 and 55. On referring to the table given in Vol. II., p. 357, we find that a degree of longitude is equal to 34:41 geographical miles under the 55th parallel, and to 30-90 geographical miles under the 59th parallel. These are represented respectively by the straight lines ab, cd in the diagonal scale. Now along the straight line drawn through the point 55, which we will call e, set off e g, eh, on either side of A B, each equal to half a b, and along the line drawn through the point 59, or f, set off fk, fl, each equal to half cd, and join kg, h; the four-sided figure, kghl, thus formed is the projection of half a degree of latitude on each side of the straight line A B, which represents the 4th meridian west from Greenwich. To extend this on either side, extend the compasses from k to h, or from 1 to g; and with the radius thus obtained, from the centre e strike the arc m n, and from ƒ as centre strike the arc op; then from e as centre, with a radius equal to ab or gh, draw arcs intersecting the arc op in o and p; and

as centre, with
a radius mf or
c d.
Join q s,

and produce it
till it meets the
straight lines
GH, DE, the
limits of the map
on the north and
south. This re-
presents the 6th
meridian of west
longitude. The
pairs of points
ux, rt, vy,
through which
the meridians
numbered 7, 2,
and 1 are drawn
respectively,
must all be de-
termined by a
repetition of the
same process.
The parallels
of 59° and 55°
north latitude,
or rather the
lines that repre-
sent them, may
now be formed
by joining the
points u, q, m,f,
n, r, v for the
former, and x, s,
o, e, p, t, y for
the latter, ruling
short straight
lines from point
to point. These
lines may be
extended as far
as requisite east
and west by
drawing faintly
an extra meri-
dian on either
side beyond the

border of the map, by the process already described for the others. The lines ux, qs, mo, np, rt, and vy must then be divided into four equal spaces like ef, and the parallels of 58°, 57°, and 56° traced by joining the points of division by ruling short straight lines from one to the other, as in the case of the parallels of 59° and 55°.

Nothing now remains but to complete the border and number the parallels and meridians, as in the projection of the map of Europe in Vol. II., p. 356. A suitable space should also be

left in the right-hand upper corner for the title of the map and a scale of miles.

We have now taken our readers through the history of the progress of geographical discovery from the earliest times to the present day, and we have described, as fully as space would allow, the political and physical features of each of the six great divisions of the world in succession. The learner has also been put into possession of the necessary knowledge for delineating and mapping out any part of the earth's surface. Much more might be said about every separate state or colony that exists in each of the great divisions, but this has been rendered unnecessary by the introduction of our "Natural History of Commerce, in which will be found all that a learner really requires about the soil, capabilities, and productions of every country of importance, whether large or small, both in the Old World and the New-the animals that are found in it, the plants that grow in it, its climate, and the influences that have rendered it hot or cold, genial or bleak, a region rich in vegetation or a

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57.-REMARKS ON THE PERSONAL PRONOUNS. (1.) The genitive mein, dein, sein are the earlier forms. The others (meiner, deiner, seiner) are the ones now commonly used. (2.) When construed with the prepositions halben, wegen, and um-willen (signifying "for the sake of, on account of "), these genitives are united with the preposition by the euphonic letters Thus meinetwegen, on et, or (in case of unser and euer) simply t.

:

barren wilderness.

LESSONS IN GERMAN. LVIII.

§ 55.-TABLE OF THE PRONOUNS.

PERSONAL PRONOUNS.

Singular.
Ich, I.

POSSESSIVE PRONOUNS.

Plural.

Wir, we.

Singular.
Mein, my.

Du, thou.

3hr, ye.

Dein, thy.

Gr, he.

Sie, they.

Sein, his.

Sie, she.

Sie, they.

Ihr, her.

6, it.

Sie, they.

Sein, its.

INDEFINITE PRONOUNS.

Man, one, a certain one.
Jemand, some one, somebody.
Niemand, no one, nobody.
Jedermann, every one, everybody.

DEMONSTRATIVE PRONOUNS.

Dieser, this.

Jener, that.

Der, this or that.

RELATIVE PRONOUNS.

Welcher, who, which.

Der, that.

Plural.
Unser, our.
Guer, your.
Ihr, their.

REFLECTIVE AND RECIPROCAL
PRONOUNS.

Sich, himself, herself, itself,
themselves.

Einander, one another.

DETERMINATIVE PRONOUNS.

Der, that, that one, he.
Derjenige, that, that person.
Derselbe, the same.
Selbiger, the same.
Solcher, such.

INTERROGATIVE PRONOUNS.
Wer? who? Was? what?
Welcher? who? which?

Wer, who, he or she, who or that. Was für? what sort of ?

§ 56. PERSONAL PRONOUNS.

There are five personal pronouns; namely, ich (I), which represents the speaker, and is therefore of the first person; bu (thou), which represents the hearer, or person spoken to, and is therefore of the second person; and er (he), fie (she), es (it), representing the persons or things spoken of, and therefore of the third person. They are declined thus:

Singular.

Plural.

FIRST PERSON, ALL GENDERS.

Nom. 3ch, I.

Gen. Meiner, mein, of me.

Dat. Mir, to me.

Acc. Mich, me.

Nom. Du, thou.

Gen. Deiner or bein, of thee.

Wir, we.
Unser, of us.
Uns, to us.

SECOND PERSON, ALL GENDERS.

Uns, us.

Ihr, ye or you.

Guer, of you.

Acc. Dich, thee.

Euch, to you.
Euch, you.

THIRD PERSON,

MASCULINE.

Nom. Gr, he.

Dat. Dir, to thee.

Gen. Seiner or sein, of him.

Sie, they.
Ihrer, of them.
Ihnen, to them.
Sic, them.

THIRD PERSON, FEMININE.

Dat. 3hm, to him.

Acc. 3hn, him.

Nom. Sie, she.

Gen. Ihrer, of her.

Dat. 3br. to her.

Acc. Sie, her.

Sie, they.

Ihrer, of them.
Ihnen, to them.
Sie, them.

account of me; um unsertwillen, on account of us, etc.
(3.) The personal pronouns of the third person, when they re-
present things without life, are seldom, if ever, used in the
dative, and never in the genitive. In such instances the corre
sponding case of the demonstrative der, die, das is employed;
thus, dessen (of this), instead of seiner; and deren (of these), instead
of ihrer.

(4.) The word selbst or selber (self, selves) may, also, for the sake of greater clearness or emphasis, be added not only to the pronouns, but even to nouns. Thus, Ich selbst, I myself; die Leute selbst, the people themselves.

(5.) Here, too, observe that the personal pronouns have also, in the plural, a reciprocal force. Thus fie lieben sich, they love one another. But as fie lieben sich, for example, might signify they love themselves," the Germans also use the word ein anter (one another), about which there can be no mistake; as, sie lieben einander. [§ 60 (3).]

(6.) In polite conversation, the Germans use the third person plural where we use the second. Thus, Ich habe Sie gesehen, I have seen you. To prevent misconception, the pronouns thus used are written with a capital letter; as, ich danke Ihnen, I thank (them) you. A similar sacrifice of grammar to (supposed) courtesy may be found in our own language, for we constantly use the plural for the singular; thus, "How are you?" instead of "How art thou?" The Germans proceed just one step beyond this, and besides taking the plural for the singular, take the third person for the second. With them, our familiar salutation "How do you do?" would be, "How do they do?"

(7.) It must be observed, however, that the second person singular, Du, is always, as in English, used in addressing the Supreme Being. It is also the proper mode of address among warm friends and near relatives. The second person plural is employed by superiors to their inferiors. The third person singular (er, fie) was used in the same manner, that is, by masters to servants, etc.

(8.) The neuter pronoun (es) of the third person singular, like the words it and there in English, is often employed as a nominative both before and after verbs, singular and plural, as a mere expletive; that is, more for the purpose of aiding the In this use, moreover, it sound than the sense of the sentence. Thus, Gs ist der Mann, is construed with words of all genders. it is the man; Es ist die Frau, it is the woman; Gs find Männer, they are men; Es donnert, it thunders; Es folgten viele, there followed many, etc.

When c8 is thus used with a personal pronoun, the arrangement of the words is precisely the reverse of the English; as, Ich bin es, it is I; Du bist es, it is thou; Sie sind es, it is they, etc.

58.-POSSESSIVE PRONOUNS

(1.) The possessive pronouns are derived, each respectively,
from the genitive case of the personal pronouns.
Note that in declining unfer and euer, the e before r is often
struck out; thus,

unsre (for unsere),
eure (for euere),

unsres (for unseres), etc.
eures (for eueres), otc.

Unfrer (for unserer), Eurer (for euerer), (2.) By their forms, therefore, these pronouns indicate the person and number of the nouns which they represent; that is, the person and number of the possessors. As, moreover, they may be declined like adjectives, they also make known by their terminations the gender, number, and case of the nouns with which they stand connected; for, in respect to inflection, a possessive pronoun agrees in gender, number, and case, not with the possessor, but with the name of the thing possessed.

(3.) The possessive pronouns, when conjunctive-that is, when joined with a noun-are inflected after the old form of declen

235

the individuals constituting a plural subject as acting one upon another, the pronoun is said to be recipro al; thus, sie beschimpfen fich, they disgrace one another.

(3.) But as (for example) fie beschimpfen sich may equally mean, "they disgrace themselves," the reciprocal word einander (one another) is added to or substituted for fich, wherever there is danger of mistake; as, fie verstehen sich einander, or sie verstehen ein. ander, they understand one another.

(4.) In the dative and accusative (singular and plural) the German affords a special form for the reflectives; viz., sich, himself, herself, itself, themselves. The personal pronouns, therefore, in all the oblique cases-that is, all cases except the nominative are used in a reflective sense, except in the dative and accusative (third person), where, instead of ihm, ihn, ihr, xc., the word fich is employed. Regarded as reflectives, the personal pronouns are declined thus :

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LESSONS IN GERMAN.

sion, except in three places (nom. sing. masc. and nom. and acc. neuter), in which the terminations are wholly omitted; thus:Singular.

MASC.

Nom. Mein,

Gen.

FEM.

NEUT.

meine,

mein.

Meines,

meiner,

meines.

meiner,

meinem.
mein.

Dat. Meinem,
Acc. Meinen, meine,

Plural.

FOR ALL GENDERS.

meine, my. meiner, of my. meinen, to my. meine, my.

(4.) When, however, these pronouns are absolute-that is, when they stand alone, agreeing with a noun understood and demanding a special emphasis-the terminations proper to the three places noted above are of course affixed; thus, dieser Hut ist meiner, nicht deiner, this hat is mine, not thine; tieses Buch ist meine, this book is mine.

(5.) But when a possessive pronoun absolute is preceded by the definite article, it then follows the new form of declension. (6.) Often, too, in this case, the syllable ig is inserted, but without any change of meaning.

(7.) When, finally, a possessive pronoun is employed as a predicate, and merely denotes possession, without special emphasis, it is not inflected at all; thus, der Garten ist mein, the garden is mine; die Stube ist dein, the room is thine; das Haus ist fin, the house is his.

(8.) It should be added that the Germans, when no obscurity is likely to grow out of it, often omit the possessive pronoun, where in English it would be used; the definite article seeming sufficiently to supply its place; as:-Ich habe es in den Händen, I have it in the hands; that is, I have it in my hands.

Singular.

FIRST PERSON.

Plural.

Unfer, of ourselves.
Uns, to ourselves.
Uns, ourselves.

Euer, of yourselves.
Guch, to yourselves.
Guch, yourselves.

THIRD PERSON MASCULINE.

Ihrer, of themselves.
Sich, to themselves.
Sich, themselves.

Gen.

Meiner, of myself.

Dat.

Mir, to myself.

Acc.

Mich, myself.

SECOND PERSON.

Gen.

Deiner, of thyself.

Dat.

Dir, to thyself.

Acc.

Dich, thyself.

Gen. Seiner, of himself.
Dat. Sich, to himself.

Acc.

Sich, himself.

(1.) Pronouns employed to represent persons and things in a general way, without reference to particular individuals, are called indefinite pronouns. Such are these

THIRD PERSON FEMININE.

Gen.

Ihrer, of herself.

Dat.

Sich, to herself.

Ran, one, a certain one.

Niemand, no one, nobody.

Acc.

Sich, herself.

Jedermann, every one, everybody.

THIRD PERSON NEUTER.

Gen. Seiner, of itself.
Dat. Sich, to itself.

Acc.

Sic, itself.

§ 59.-INDEFINITE PRONOUNS.

Jemand, some one, somebody.

The following, which also belong to this list of indefinites,

have already been treated of under the head of indefinite nume

rals; viz. :

Gtwas, something.

Rits, nothing.

Seiner, no one, none.

Finer, one, some one.

Jae, each, every one.

Janeer, each, every one.

Jeglicher, each, every one.

Einige, somewhat, some.

Etliche, some, many.

Aller, every one, all.

Ihrer, of themselves.
Sich, to themselves.
Sich, themselves.

Ihrer, of themselves.
Sich, to themselves.
Sich, themselves.

KEY TO EXERCISES IN LESSONS IN GERMAN.
EXERCISE 153 (Vol. III., page 139).

1. Viele Leute bringen ihre Zeit in Trägheit zu. 2. Er brachte den Mancher, many a, many, seve- größten Theil seines Lebens in fremden Ländern zu. 3. Jeder Mensch,

ral.

Jebermann is declined thus :Nom. Jebermann, everybody. Dat. Jedermann, to everybody. Gen. Jedermanns, of everybody. | Acc. Jedermann, everybody. (2.) The German man (like the French on) is used to indicate persons in the most general manner; thus, man sagt, one says; that is, they say, people say, it is said, etc. It is indeclinable, and is found only in the nominative; when, therefore, any other case would be called for, the corresponding oblique case of ein is employed; thus, er will einen nie hören, he will never listen to one, i.e., to any one.

(3.) Jemand and Niemand are declined alike; thus:

Singular.

Nom. Jemand, somebody.

Gen. Jemands or Jemandes, of somebody.
Dat. Jemand or Jemandem, to somebody.
Acc. Jemand or Jemanden, somebody.

(4.) Note that the second form of the dative (Jemandem, Nie manten) is seldom employed except when the other form would leave the meaning ambiguous. Thus, es ist Niemandem nüßlich, it is useful to nobody; where, were Niemand used, the sense might be, nobody is useful. This remark applies also to the accusative; as, fie liebt Niemanden, she loves nobody: in which instance, were the other form (Niemand) substituted, it might mean, nobody loves her.

00.-REFLECTIVE AND RECIPROCAL PRONOUNS. (1) When the subject and the object of a verb are identical, the latter being a personal pronoun, the pronoun is said to be reflective, because the action is thereby represented as reverting upon the actor; thus, er rühmt sich, he praises himself.

(2.) When, however, in such case, the design is to represent

welcher Gefühl für Ehre hat, entzieht sich keinen Pflichten, welche der Menschheit Nußen bringen. 4. Er sagt, sein größter Schaß sei Gott, und die ganze Welt, mit Ihm verglichen, sei nichts. 5. Dieser Mann sagte, es sei ihm ganz einerlei, ob seine Unternehmungen erfolgreich wären oder nicht. 6. Wie viele Sorten Wein haben Sie? 7. Ich habe dreierlei, Sie mögen wählen, welche Sie wollen. 8. Ich gehe jeden Tag zweimal über die Londoner Brücke. 9. Viele gehen über Oftende nach Deutschland. 10. Ich werde wahrscheinlich einen Monat in Bonn zubringen. 11. Mein Nachbar hat treierlei Enten in seinem Teiche; dieselben sind sehr schön. 12. Dreierlei Rosen wachsen in unserm Garten. 13. Wenn ich hungrig bin, so ist es mir einerlei, ob ich Wildpret oder ein Stück Rinderbraten vor mir habe. 14. Er kaufte Band von dreierlei Farben.

EXERCISE 154 (Vol. III., page 139).

1. This year the fruits of the garden, as well as of the field, have turned out well. 2. This tree yields abundance of fruit every year. 3. Is all produce fruit? 4. No, not all, but only that which grows on trees. 5. This young man relies too much upon his relations, and too little upon his own abilities. 6. He depends upon our visiting him next week. 7. He trusted that God would help him. 8. He who relies too much upon others, may easily be deceived. 9. I highly esteem my friends. 10. He thinks much of a comfortable life. 11. This man thinks too much of himself and his prudence, therefore he despises the counsel of well-wishing friends. 12. Only upon this

condition can I agree to it. 13. I agree to it if it has no evil consequences.

14. He agreed to it without being acquainted with all the difficulties. 15. This child acts just as if it were at home here. 16. The sailor acted as if he were out of his senses. 17. He behaves as though the greatest wrong had befallen him. 18. This man behaves as though he were offended. 19. He acts like a child of five years of age. 20. The neighbour thrust the intruder out of doors. 21. The boy rushed out at the door as I opened it. 22. You who quarrel, go out of doors (Goethe). 23. It entirely depends upon circumstances whether I start for America next year or not. 24. It greatly depends upon circumstances what he will do. 25. The peasants lead as independent a life in America as in Germany they

LESSONS IN MENSURATION.-IV.

AREAS OF IRREGULAR FIGURES AND FIGURES BOUNDED BY CURVED LINES.

PROBLEM XII.-To find the area of a regular polygon, the length of the side being given.

Rule: Find the radius of the inscribed circle by previous rule; then multiply the length of the side by the number of sides, and this by the radius, and half the product will give the area. EXAMPLE.-The length of the side of a regular pentagon is 3; what is its area?

Find the radius of the circumscribed circle, thus: 360° 72° The (angle) at the centre = = 72°; 5 2 AOC (Fig. 15, page 77). 3

= 36° =

Then half the side or = 15 = base of right-angled triangle

or

base 1.5 AO C, and hypothenuse ▲ O = =2.55. nat. sine 36° -5878 Again, perpendicular oc = hypothenuse X nat. sine OAC, or 2.55 x 3090 = 2·06 = radius of inscribed circle. 3 x 5 x 2.06 Then 2

15:45, area of pentagon. EXERCISE 16.

1. What is the area of a pentagon whose side is 3.82 ? 2. The side of a hexagon measures 20 poles; what is its area? 3. The side of an octagon measures 20 yards; what is its area? 4. The side of an equilateral triangle is 389 links; required its area.

5. The side of an octagon is 156 feet; what is its area? PROBLEM XIII.-To find the area of any irregular figure, the boundary sides of which are straight.

Rule: Divide the figure into separate triangles. If the diagonals are given, find at once the area of the respective triangles, as explained in Problem XI., and their sum will be the If the diagonals are not given, they must be obtained by actual measurement.

area.

EXERCISE 17.

1. The four sides of an irregular figure are as follow:BA= 12; AC = 20; CD=18; and D B = 10; and the diagonal from A to D measures 6. What is the area of the figure?

2. The four sides being as above, but the diagonal being 12, what is the area?

3. A figure has five sides, as follows:-AB = 22; BC=18: CD = 32; DE = 18; and E A = 20; and the diagonals E B and BD measure respectively 23-25 and 16.75. What is the area of the figure?

We shall now consider the superficial area of surfaces bounded by curved lines; and we request our reader to refresh his memory by a reference to our remarks upon the proportion which exists between the diameter and the circumference of a circle.

PROBLEM XIV.-The radius of a circle being given, to find its area.

Rule 1: Multiply the radius by the circumference, and halve the product.

Note. The circumference of a circle being to its diameter in the proportion of 3.1416 (approximately) to 1, it follows that its proportion to the radius is as 31416 (a number we shall designate generally by ) to, and hence the truth of the

above rule.

EXAMPLE 1.-Required the area of a circle whose diameter (D) is 1. D 1 Here R=

or

1 π 3.1416

=7854.

2; and the area is 2 X 2 or 4 This number may be with great advantage borne in mind by the student, it being the area of a circle whose diameter is unity. It is often used in estimating circular areas.

lead a dependent one. 26. No man on earth can become quite independent.

EXERCISE 155 (Vol. III., page 139).

1. Voriges Jahr sind die Früchte nicht gut gerathen. Dieser Baum trägt nur selten Früchte. 3. Dieser junge Herr verläßt sich zu viel auf seine Fähigkeiten. 4. Nein, er verläßt sich nicht zu viel auf seine Fähigkeiten, denn er weiß, daß es nicht gut ist, sich auf diejenigen Anderer zu verlassen. 5. Ich verlasse mich auf Sie, daß Sie mich nächste Woche besuchen werten. 6. Thun Sie gerare, als ob Sie zu Hause wären. 7. Der Verbrecher stellte sich, als ob er wahnsinnig wäre. 8. Dieser Mann stellt sich gerade wie ein Kind. 9. Wo ist Ihr Kanarienvogel? Er ist zum Fenster hinaus geflogen. 10. Wie kann ich in eine Sache einwilligen, die gegen meine Neigung ist? 11. Ein jeder, der sich entzweit, wird aus dem Hause getrieben. 12. Es hängt von Umständen ab, ob ich zu meinen Freunden gehen werde. 13. Jeder Mensch strebt unabhängig zu werden. 14. Ver Lasse vich darauf, daß ich dir nicht wieder helfen werde

EXERCISE 156 (Vol. III., page 179).

1. It is not your fault that you are so unhappy. 2. It was not his fault that he broke this glass. 3. I can give nothing for it, except my thanks. 4. I shall state the reasons for it, if it be requested. 5. Can you tell me what o'clock it is ? 6. No, for my watch has stopped. 7. Has your watch stopped long? 8. Yes, nearly an hour. 9. My watch goes too fast, it has gained nearly half an hour. 10. My friend's watch is five minutes too fast. 11. Good-bye, and do not forget to visit me soon again. 12. Good-bye, sir. 13. When shall we both visit Mr. N.? 14. It depends entirely upon you what time you wish to appoint for it; I am ready at any time to accompany you. 15. It depends upon you to save or to ruin this family. 16. The neighbour works in his garden, and tries to put it in order. 17. With all his exertions he never settles this matter. 18. He tried to get me into the ranks of his comrades. 19. It is difficult to accustom a disorderly man to regularity. 20. After great trouble he has cleared up the account. 21. He who stops at the foot of a steep mountain, and through fear of exertion omits to climb up it, and prefers resigning a beautiful view, shows thereby that he is a weak person, and unworthy of such enjoyment; and he who stops in the midst of his intellectual cultivation through his own fault, and is willing to dispense with the sweet kernel of wisdom because a rough and hard shell surrounds it, likewise shows not only his unworthiness to enjoy it, but also how little he has understood the vocation and the duty of man as a spiritual being.

EXERCISE 157 (Vol. III., page 179).

1. Ich kann nicht dafür, daß Sie das Unglück gehabt haben. 2. Sie konnten nicht dafür, daß die Magd den Teller zerbrochen hat. 3. Er konnte mir nichts dafür geben, als seinen Dank. 4. Er konnte nichts dafür, er sprach nur die Wahrheit. 5. Kann der Kutscher etwas dafür, daß der Wagen umgeworfen wurde? 6. Nein, er konnte nichts dafür, denn die Pferde waren nicht zu beruhigen. 7. Können Sie mir sagen, welche Zeit es ift? 8. Nein, meine Uhr geht nach. 9. Die Stunde meiner Abreise zu bestimmen hängt von meinen Eltern ab. 10. Leben Sie wohl, Madam; und vergeffen Sie nicht, mich Ihren Eltern zu empfehlen. 11. Es hängt von Ihnen ab, welche Zeit Sie bestimmen wollen, Ihre Freunde zu besuchen; ich werde stets bereit sein, Sie zu begleiten. 12. Glück und Unglück, Leben und Tov, Armuth und Reichthum, Alles hängt von dem Willen Gottes ab. EXERCISE 158 (Vol. III., page 179).

1. The thief was convicted of his crime, and of course he will be punished. 2. The father went away this morning, and has not yet returned. 3. The book has been lost, and all these scholars pretend not to know where it has gone to. 4. My nephews went away without saying where they were going. 5. Our fruit is all gone. 6. Any amount of money will go if one is wasteful. 7. The Turkish emperor, Soliman II., said, shortly before his death, "My strength is gone, but not my courage." 8.. How far are you going to walk ? 9. I walk till I get tired, generally as far as the park. 10. My friend knows very well how far he has to go in this affair. 11. Even in joke one ought to know how far one can go, because even in jest one may offend. 12. Where are you going? 13. I am going to my attorney. 14. How far have you to go? 15. To the end of the town. 16. How

long will it take you to walk ? 17. More than an hour. 18. How

far have you walked? 19. I have been as far as the river. 20. How long have you been walking? 21. I have been walking above half an hour. 22. How long have you been from home? 23. I have

been away three-quarters of an hour. 24. Have you been far away from it? 25. I have been nearly half an hour's walk from home. 26. I hope to see you again, whether it be in this world or in the next. 27. The prisoner thought it was now long enough for him to have been obliged to dispense with the warm light of the sun and the fresh air. 28. I cannot come to your house to-morrow, unless my brother should quite recover till then. 29. I cannot possibly finish this letter to-day, unless I am less disturbed this afternoon. 30. Nobody will be admitted into the town, unless he has a passport.

EXAMPLE 2.-The radius of a circle is 1. What is its area?
In this case, R being 1, D is 2, and the circumference becomes
2 or 6.2832.
6.2832
Therefore the area is R X
- 3.1416.
2
EXERCISE 18.

1. The diameter of a circle is 3; what is its area?
2. The circumference of a circle is 3.1416; what is its area?
3. The diameter of a circle is 4 ft. 11 in.; what is its area?

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