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of orange peel is squeezed between the fingers. Although the characteristic of agreeable odour is a very good common sign by which we may be justified in expecting that a plant may, in certain cases, belong to the Myrtaceae, nevertheless it is a very loose sign when taken apart from others. We can only arrive at a correct botanical comprehension of the Myrtacea by studying some of the generic characters that have been mentioned in our preceding list.

If the specimen of common myrtle under examination be a sprig, not a single leaf, the student, before he lays it down, should observe that the leaves are opposite, not alternate (Fig. 143). Let us now examine the flower. The particular species under consideration has a calyx of five divisions, and there are also five petals, but in certain species these floral parts are generally four. The stamens are numerous, as will be readily observed; and the reader need not be told at this period of our labours that it is necessary to ascertain whether these stamens grow from the calyx or the receptacle. They grow from the calyx, as will be readily distinguished. The ovary is inferior; it contains three little cells, and each cell contains many ovules; and it shoots up a single style, which terminates in a small stigma so very minute that it cannot be seen by the naked eye.

Figs. 141 and 142 are representations of a vertical section of the flower and ovary of a common myrtle, and a transverse section of the ovary with adherent calyx, or rather the fruit with adherent calyx. If the reader examines Fig. 142, he will observe that the number of seed-cells in the species of myrtle under consideration is three, or, to use the language of Botany, the ovary is trilocular, or three-celled. If the reader now refer to the list of characteristics of this family, he will find the expression, "ovary, usually two to six-celled," which signifies that the number of cells may vary between two and six.

By well considering the characteristics already discussed, the student will be at no loss to recognise an individual of the myrtle tribe, even without taking into consideration minute microscopic peculiarities.

Let us now proceed to mention a few particulars in connection with the dimensions, natural habitation, and properties of this beautiful and useful botanical order.

The stem of the Myrtacea is generally woody, the leaves opposite or alternate, simple, entire, rarely stipulated; frequently, as we have seen, provided with secretive glandular appendages, embedded in the parenchyma. The flowers are complete, regular, solitary, or irregularly agglomerated. The greater number of myrtaceous plants have berries for their fruits; but some others, the principal of them being Australian plants, have a dry hard fruit; these, too, have alternate leaves, which is not usual in the myrtle tribe. The great districts for myrtles are the intertropical regions and Australia; only a few species existing in temperate climes. The spice sold as cloves is the produce of one of the myrtle tribe, Caryophyllus Aromaticus (Fig. 144), of which cloves are the dry flower-buds. Allspice is the berry of another (Eugenia Pimenta). Guava jelly, so valued and esteemed wherever it can be procured, is the conserve made of the mashed berries of a myrtle which grows in the West Indies. The pomegranate, too, a native of Northern Africa, but which

now grows in the south of Europe, furnishes another example of a fruit-bearing myrtle.

In reviewing, then, the chemical and physiological characteristics of the Myrtacea, we learn that none of the tribe are poisonous. The greater number contain an abundance of fragrant oil. Some yield fruits which are delicious to eat; and all are imbued with a certain, but variable amount of astringent matter, similar to that contained in oak bark, whence it has been denominated tannic acid by the chemist.

In this country the myrtle requires protection during the

winter months in all districts north of the Thames, but in the the south of England, and especially along the coast of the western counties, it flourishes against a wall, often growing to a great height, and covering a considerable space with masses of dark-green foliage.

Many of the species of the myrtle tribe are very large trees. The Sapucaya tree, as it is called in Brazil (Lecythis Ollaria), is of the tallest trees amongst the very tall ones that grow in n forests. In Fig. 136 the reader will find a representabranch of this species. How different from a branch nmon myrtle!

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LESSONS IN GERMAN.—XXIV.

SECTION XLV. PECULIAR IDIOMS.

Sich erinnern corresponds, in signification, to the English verb 'remember," as:-Er erinnert sich meiner (§ 126), he remembers me (literally, he remembers himself of me). Ich erinnere mich jener schönen Zeit, I remember (remind myself of) that sweet time.

In referring to a thing already learned, the verb behalten, "to keep, to retain," is generally used, as :-Ich kann die Wörter nicht (im Gedächtniß) behalten, I cannot retain or remember (keep in memory) the words. Er macht so viele Fehler, weil er die Regeln nicht behält, he makes so many mistakes, because he does not retain (remember) the rules.

1. The dative of a personal pronoun is frequently used instead of a possessive pronoun, as::-Ich habe mir den Finger abgeschnitten, I have cut off my finger (I have to me the finger cut off). Gr gab es mir in die Hände, he gave it into my hands (he gave it to me in the hands).

2. The phrases es fällt schwer or es hält schwer are nearly synony mous, and signify "to be hard, to be difficult," as:-Diesem armen Manne fällt es schwer, zu betteln, it is hard (it comes hard) for this poor man to beg. Es hielt schwer, ihn zu beruhigen, it was difficult to calm him.

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ja, ich erinnere mich desselben noch sehr gut. 1. Können Sie sich des Tages meiner Ankunft nicht mehr erinnern? 2. 3. Es giebt viele Menschen, rie sich lieber ihrer schlechten, als ihrer guten Thaten erinnern. 4. Ge bielt schwer, ibn von ter Wahrheit richer Erzählung zu überzeugen. 5. Ge rat zuweilen schwer, etwas zu glauben, was wir nicht begreifen können. 6. Es Anfrruch nehmen zu müssen fällt dem armen, aber fleißigen Manne schwer, die Hülfe fremter Leute in 7. Die englische Srrache fällt mir schwerer, Kenntnissen fiel es ihm nicht schwer, eine eben so gute, als angenehme Steile als die französische. 8. Vei seinem Gelte, seinen Verwandten und seinen

zu finden. 9. Warum fernt Ihr Bruter so viel schneller, als Sie? 10. Weil er ein besseres Gedächtniß hat, und die Werter besser behalten kann. 11. Können Sie sich nicht mehr erinnern, wem Sie die Bücher und ros Barier gegeben haben? 12. Ich kann mich dessen nicht mehr erinnern, 13. Der Faulenzer behält die Regeln nicht, weil er dieselben nicht grüntisch 15. Gr trudte tem armen Manne einen Thaler in die Hant. 16. In lernt und lieber spielt, als arbeitet. 14. Der Diebstahl ist ein Verbrechen. der Ferne erinnern wir uns gern ter Freunte. 17. Junge Leute tragen zuweilen Brillen, um gelehrt auszusehen.

EXERCISE 87.

1. Do you remember the day of the arrival of your friend? 2. Yes, I remember the day very well. 3. Most people remem ber the years of their youth with pleasure. 4. There are many who remember their follies with shame. 5. It is difficult to remember every rule of a language. 6. It is not so difficult to convince a learned as an unlearned man. 7. Is it difficult to submit to the adversities of life? 8. Yes, it is very difficult; but the thinking man conquers them. 9. Can you not remem ber to whom you have lent my bok? 10. No, I cannot remem ber. 11. To lie is a sin.

SECTION XLVI.-VERBS GOVERNING THE GENITIVE.

Some verbs in German govern the genitive (§ 125), while those in English of corresponding signification require the objective, as-Gedenke meiner, remember me (or, think of me). Ich achte seiner nicht, I do not regard (notice) him. Er bedarf des Geldes, he needs (wants) money. Er tenkt or getenkt meiner, he thinks of me. Gr erwähnte unser, he mentioned (spoke of) us. Er beraubte ihn all seiner Schäße, he robbed him of all his treasures. Das Haus entbehrt des Vaters, the house lacks (misses) the father (master). Der Unglückliche harrt besserer Zeiten, the unfortunate (waits for) expects better times. Schonet mein (§ 57. 1), spare me. Sie frotten meiner, Pring, you mock me, prince! Vergesset meiner nicht, forget me not. Er bediente sich der besten Mittel, he used (served himself of) the best means.

Some verbs of the above class (§ 125) more commonly take the accusative, as :-Verziß deine Bücher nicht, do not forget your books.

Alterschwäche, f. de-
crepitude.
Beürfen, to need,
want.

Betim mern, to grieve, trouble.

Bereuen, to repent,
regret.
Beheiten, modest.
Bitte, f. request, pe-
tition.

Californien, n. Cali-
fornia.

Cholera, f. cholera.

Dann, then.

Dumm, stupid.

Entlich, at length.

Entbehren, to be in

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Harren, to hope, wait
for.
Hierfür, for this, for
it.

Jammern, to distress,
grieve, lament.
Längst, long since
(ago).

Regiment', n. regi

ment.

Sehnsucht, f. longing.
Schonen, to spare.
Spotten, to mock,
scoff at.
Sterben, to die.
Strenge, severely.
Ueberstehen, to over-
come, endure.
Unentbehrlich, indis-
pensable.

insuffer

Unleid'lich,
Leid, n. pain, sorrow. able.
Lohnen, to reward. Un'wahrheit, f. false-
Nachlässigkeit, f. neg-
hood.
ligence, remiss- Verweisen, to repri-

ness.

mand.

der Straße läuft. 13. Wenn den Fürsten das Volk jammerte, so würde er
anders regieren. 14. Aber das Volk wird ihm hierfür noch lohnen, und
dann seiner nicht schonen. 15. Es könnte wohl der Mühe lohnen, nach
Californien zu reisen. 16. Ich würde gern diese Kleiter schenen, wenn ich
antere hätte. 17. Ich wünsche keine andern Tov zu sterben, als den Tod
vor Altersschwäche. 18. Vergiß meine Worte nicht. 19. Vergiß der
20. Wenn
überstantenen Leiren, aber vergiß nicht die genossenen Freuden.
rie Fürsten könnten, so schonten sie weder der Freiheit, noch sonst eines
Rechtes ihrer Völker.
EXERCISE 89.

1. She nursed her father in his old age, and nursed me when
I had the nervous fever. 2. He mocked me, but observed not
how the people mocked him. 3. Has he accepted my present?
4. No, he told me he needed not the present. 5. Do not
mention his kindness. 6. The teacher dares not spare the
negligence or falsehood of his scholars, but must reprimand
them severely when he observes it. 7. Forget not the warning
voice of your parents. 8. Remember the Sabbath-day.
Who can believe a man who sneers at everything and scoffs at
everybody? 10. We waited with longing for the arrival of our
friends. 11. When thou repentest of thy faults, then shall I
remember thee with pleasure. 12. Conscientious people make
no vain speeches, nor make parade of qualifications which they
do not possess.

9.

Appended to this and subsequent lessons the student will find a Key to the Exercises in German. Our reasons for not beginning this key before are the same as those which we have given for not commencing the Key to Exercises in Lessons in French at an earlier period.

KEY TO EXERCISES IN LESSONS IN GERMAN. Exercises 1 and 2 are on Pronunciation. Exercise 3 is on Handwriting. EXERCISE 4 (Vol. I., page 38).

1. Who has bread? 2. The baker has bread. 3. Has the baker flour? 4. Yes, he has flour also. 5. What has the miller? 6. The miller

want of, dispense Rete, f. speech, Wahrnehmen, to ob- has flour and grain. 7. Who has meat? 8. The butcher has meat.

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9. Have you beer? 10. No, the brewer has beer. 11. Have you wine? 12. No, I have coffee. 13. What has the girl? 14. The girl has tea. 15. Has the brewer grain ? 16. No, he has only beer and wine. 17. What has the child? 18. It has water. 19. Has it bread also ? 20. Yes, it has bread and meat also.

EXERCISE 5 (Vol. I., page 38).

1. Do you love the child or the man? 2. I love the child. 3. Have 5. Does the child you the sugar? 4. No, the child has the sugar. love the girl? 6. Yes, and the girl loves the child. 7. Who has the glass ? 8. The child has the glass. 9. Has the brewer the wagon? 10. No, the peasant has the wagon. 11. Who has the beer? 12. The

brewer has the beer and the wine. 13. Has the miller the flour or the

bread? 14. He has the flour. 15. Has the baker the wine or the water? 16. He has the water. 17. Do you love the peasant? 18. No, I love the teacher. 19. Have you meat or wine? 20. I have the meat. 21. Have you the bread or the sugar? 22. I have the bread.

23. Has the father the book or the comb? 24. He has the book.

EXERCISE 6 (Vol. I., page 61).

Enjoy (the) life, but with
He was accused of a crime that 3. What has the butcher's friend? 4. He has the peasant's dog and

1. Are you the baker's friend? 2. No, I am the joiner's friend.

honour!

he had not committed. Do not forget me.

flagt, tas er nicht began'gen hatte. Bergessen Sie meiner nicht. Ich vergaß meinen Bleistift; geben I forgot my pencil; give me

Sie mir einen Augenblick den

Ihrigen.

yours a moment.

EXERCISE 88.

horse. 5. Where is the flour? 6. It is in the miller's bag. 7. Where is the grain? 8. It is in the peasant's basket. 9. Who loves the teacher ? 10. The scholar loves the teacher. 11. Are you sleepy? 12. No, I am thirsty. 13. Where is the brother's ball ? 14. The child has the brother's ball in the father's hat. 15. Where is the teacher's horse? 16. It is in the stable. 17. Does the joiner praise the carpenter? 18. No, the carpenter's son praises the teacher's son. 19. Where is the joiner's chair? 20. It is in the teacher's room. 21. Does the carpenter love the teacher? 22. Yes, he loves and 23. The man is at the table, the book is on the table, and the dog is under the table.

EXERCISE 7 (Vol. I., page 62).

2. Wenn man auf jede Rete achten wollte, bätte man sich um viele Sachen praises the teacher.
1. Wer alte Leute nicht achtet, ist nicht werth, selbst geachtet zu werden.
zu bekümmern. 3. Er entbehrte der nöthigen Mittel, um seine Pläne aus
zuführen. 4. Wer wird sich meiner annehmen, wenn ich verlassen bin? 5.
Wenn er seine Fehler bereut, so will ich ihrer auch nicht mehr gedenken. 6.
Ich würte noch viel mehr Sachen bedürfen, wenn ich nicht gewohnt wäre,
die Gegenstände zu entbehren, tie (Sect. XXI. 3) viele Leute für unent
behrlich balten. 7. Der General erwähnte Ihres Sohnes, als eines der
tarfeesten Männer in seinen Regimentern. 8. Gewähre meine Bitte, D

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1. Who has this girl's paper? 2. This child has it. 3. Whose book has this scholar? 4. He has the teacher's book. 5. From whom have you this leather? 6. I have it from the shoemaker. 7. For whom is this apple? 8. It is for the saddler's child. 9. Whose coat has the tailor's scn? 10. He has this friend's ecat. 11. From whom has this hatter's son money? 12. He has money from the father. 14. The teacher's friend has it. 15. Whose house and garden has the teacher? 16. He has the

10. Nichts ist unleitlicher, als auf Jemanden lange zu warten, der zuleht 13. Where is the peasant's wagon ? gar nicht femmt. 11. Längst schon harrte ich Ihrer mit Sehnsucht, als ich Sie entlich kommen fah. 12. Erbarme tich des Kindes, das verlassen auf mayor's house and garden. 17. From whom have you this hat

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EXERCISE 8 (VOL. I., page 62).

1. Where is the mate's brother? 2. He is with the captain in the ship. 3. Is the nobleman's son with him also? 4. No, he is in Germany. 5. Where is the father? 6. He is with the captain in the custom-house. 7. Does the captain praise the nobleman's sou? 8. Yes, and he praises the father also. 9. Does the nobleman love the captain ? 10. Yes, he loves and praises him very much. 11. Is this man the captain's son? 12. No, he is the mate's son. 13. Is this sailor rich ? 14. No, he is poor and merry. 15. How old is this man? 16. He is not very old. 17. Is he sick? 18. No, he is hungry. 19. What does this girl give the child? 20. She gives it only sugar. 21. What do you give the servant? 22. I give him money. 23. What does the servant give the horse? 21. He gives it hay. 25. Does this child love the teacher ? 26. Yes, and the teacher praises the child. 27. Is the hunter still in the forest ? 28. Yes, and the nobleman's son is with him. 29. The huntsman goes to the forest to the father, and I go to the brother.

wagon

EXERCISE 9 (Vol. I., page 66).

1. Has a man or a child this friend's stick? 2. This man has an enemy's sword, and this child has a friend's stick. 3. What has the hunter ? 4. He has a dog and a gun. 5. Who has the peasant's plough? 6. The father of this child has the plough. 7. Has this blacksmith the merchant's money? 8. No, he has ouly iron from a merchant. 9. Have you the baker's wagon? 10. No, I have this from a carriage-maker. 11. Have you this girl's ribbon? 12. No, I have cloth from a draper. 13. Have you this friend's coat? 14. No, I have this coat from a tailor. 15. Have you the teacher's paper? 16. No, I have this paper from a stationer, and a letter of recommendation from the teacher. 17. Is the horse a draught animal? 18. Yes, and it is also a beast of burden. 19. Is the camel a draught-animal also? 20. No, it is only a beast of burden. 21. Whose law-book has the nobleman's son? 22. He has the law-book of the judge of the superior court.

LESSONS IN GEOMETRY.—XIV.

THE CIRCLE AND ITS PROPERTIES.

HITHERTO, in our lessons in Geometry, the attention of the student has been directed to the construction of rectilineal figures, or figures contained by right or straight lines: we shall now enter on what may be termed the "geometry of the circle," or the method of drawing circles and parts of circles under various conditions; concluding our lessons on this subject with instructions for drawing regular polygons by the aid of the circle, protractor, and scale of chords, as well as the ellipse and other figures bounded by curves or consisting of curved lines.

It may be useful to the student if we recapitulate briefly the names of various parts of the circle, and mention its chief properties as laid down in the Definitions (Vol. I., page 53), before explaining one or two other points that will be necessary for him to understand before he reads the problems that we are about to bring under his notice.

As we

Firstly, let us ask, What is a circle? It is a form that meets the eye often enough as we go about our daily tasks. pass through the streets of town or city, or along the highways and byways of the country, it is brought before us in the wheels of every vehicle we meet. It is exhibited in the form of the majority of our cooking utensils. If we turn our eyes to the face of the clock that stands on the mantel-shelf, or the watch that is carried in the waistcoat-pocket, it is there. Nay, more, it is found in every button that we wear on our attire, in the cups and glasses out of which we drink, and in the plates off which we eat our daily food. It is the most perfect, the most elegant, the most useful of all forms. Under the figure of a snake holding its tail in its mouth, the ancients adopted it as the emblem of eternity, which had no beginning, and which has no end. It is a figure which any one can describe by the aid of a pair of compasses with the greatest ease, but one which it would be most difficult to draw without the assistance of this useful instrument.

There was a man once, though, who could draw a perfect circle with a simple sweep of his unerring arm and hand, and mark its centre with the same rapidity and precision. His

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name was Giotto or Angiolatto, an Italian painter, sculptor, architect, and engineer, born at Vespignano in 1276. As a boy he was employed as a shepherd, but Cimabue, who accidentally discovered his innate talent for drawing, took him by the hand, and made him a greater painter than himself. He had learnt the rudiments of his art in the fields by sketching his sheep on the earth with the end of his shepherd's crook, or with a nail on any flat piece of stone that might come in his way. These small beginnings had great results in Giotto's case, for he went on step by step until he became the greatest Italian painter of his time. When Benedict XI. was Pope of Rome, artists were wanted to work at the decorations of the great cathedral dedicated to St. Peter, and invitations were sent to the principal painters of Italy to forward specimens of their skill for the pope's inspection. Giotto contented himself with drawing a circle on a piece of paper with a bit of charcoal, and handing it to Pope Benedict's messenger. It was in vain that the messenger urged that his master required some design as a speci men of Giotto's skill, for the painter refused to send anything else. The circle so hastily drawn was found to be perfect when tested with a pair of compasses, and so struck was the pope and his advisers with this surprising proof of the artist's capacity as a draughtsman, that he was immediately summoned to Rome to carry out the work that Benedict wished to contribute as his quota to the adornment of the finest cathedral that has yet been built.

But to return from this digression. The circle, in geometrical terms, is a plane figure; that is, a figure drawn on a plane or level surface, and bounded by a curved line called the circumference or periphery. Let us explain these terms; for there is nothing so well calculated to fix the meaning of a word and the peculiar property of the figure that it is intended to describe as to trace it to the primary source or root from which it is derived. The word plane is derived from the Latin planus, flat, smooth, level. It is merely another form of the word plain, which we apply to a level tract of country because it is flat and devoid of hills or any striking inequalities in its surface. A joiner or cabinet-maker will now see at once the reason why the tool he uses to give an even level surface to a piece of wood is called a plane. The word circumference, which is applied to the line which is carried round about, or which bounds any figure, is derived from the Latin words circum, round, and fero, I bear or carry. The word periphery means precisely the same thing, but it traces its source to the Greek instead of the Latin, being derived from the Greek Teρi (per'-ry), around, and pepw (fer'-ro), I bear or carry.

с

M

B

0

N

E

Fig. 49.

Look at the annexed figure. The whole of the superficies or surface of the paper that lies within the curved line A C B E is called a circle. The curved line AC BE itself is called the circumference or periphery of this circle. The point o is called its centre, a word derived immediately from the Latin centrum, and more remotely from the Greek Revтpov (ken'-tron), a sharp point. The position of this point has this peculiar property: it is such that all straight lines drawn from it to the circumference are equal to one another. Thus the straight lines o A, 0 B, C, o E, drawn from the centre o to the points A, B, C, E, in the cir cumference, are equal to one another. These lines are called radii of the circle A C B E, from the Latin radius, a sunbeam or ray of light, and hence applied to any line or any number of lines that radiate in various directions from the same point, as rays of light seem to proceed from the sun or any luminous centre, as may be seen by looking at a candle or gas-light with halfclosed eyes, when the rays that seem to issue from it will become distinctly visible. Any two radii that proceed from the centre in opposite directions, and therefore lie in the same straight line, form together a straight line called a diameter of the circle. In the above figure (Fig. 49), A B is a diameter of the circle ACEE Its name, derived from the Greek Sta (di'-a), through, and METPE (met'-rine), to measure, implies that it is a line that measures the circle across its superficies and through its centre. Having arrived at the meaning of the word diameter, we arrive at the full significance of the term "diametrically opposite." Thus, when we say that the opinions entertained by any two men are diametrically opposite, we mean that they are as contrary to each other as it is possible to be-as opposite, in fact, in

direction and aim as the two radii proceeding from the centre of a circle that together form a diameter of that circle.

Any portion of the circumference of a circle, no matter how great or how small, is called an arc of that circle. Thus the portion AC of the circumference of the circle A C B E is an arc of that circle. For the sake of clearness we will call this portion the arc AM C. The remainder of the circumference, C B A, is also an arc of the circle, as are also the portions A C B, B E A, E A C, C B E, etc. etc. The straight line that joins the extremities of an arc is called the chord of that arc. Thus AC is the chord of the arc AM C, AE is the chord of the arc A N E, and ▲ B is the chord of the arc A C B or the arc A E B. The student may ask why a portion of the circumference of a circle is called an arc, and to this we may reply by two counter-questions-Why is a man who shoots with a bow and arrow called an archer? and, What does the arc and its chord resemble ? It may not be easy to give a reply to our first counter-query, but it will suggest a reply to our second, namely, that the arc and its chord look very much like a bent bow and its string tightened to its utmost tension. Let it now be said that the Latin word for a bow is arcus, and the meaning of the words arc and archer becomes clear enough. The word chord is derived from the Latin chorda, the string of a harp or lyre, and hence any kind of string. It is from this word that we have "cord," a term applied to any small rope, or thick or closely-twisted string. From this it may be seen how much a letter more or less disguises an otherwise familiar word.

A segment of a circle is any part of the surface of the entire circle enclosed by an arc and its chord. The word segment is derived from the Latin segmentum, a piece, shred, or paring. Thus in Fig. 49, A M C represents a small segment of the circle ACBE contained by the arc A M C and the chord A c. In like manner, the portion of the circle C B E A, contained by the arc CBEA and the chord c A, is a segment of the circle A C B E, as is also A C B, contained by the arc ▲ C B and the chord A B. This chord, however, is a diameter of the circle A C B E; and when a diameter is the chord of an arc that encloses a segment in conjunction with it, the portion of the surface thus enclosed is called a semicircle or half-circle.

The figure enclosed between any two radii of a circle and the are of the circumference intercepted between the extremities of the radii that touch or meet the circumference, is called a sector, from sectum, a part of the Latin verb seco, to cut. Thus o c E, the portion of the circle A C B E bounded or enclosed by the radii, OC, O E, and the arc C A E, is the sector of that circle, as is also the remainder of the surface; namely, the portion bounded by the radii o C, O E, and the arc C B E.

A

We shall have occasion in the course of future problems to speak of circles touching or meeting one another, and of straight lines that are tangents to a circle. In Fig. 50, the smaller circles ADE, C F G touch the larger circle A B C, the former in the point A, and the latter in the point c. One circle can touch another in only one point. The circumference of the touching circles meet in one point only. If, however, the surface of one circle overlaps the surface of the Xother in the slightest degree, contact is destroyed, and the circumferences of the

B

Y

Fig. 50.

circles are said to cut one another. Thus in Fig. 51, the circum

R

Rs in the point c. Straight lines drawn at right angles to the
tangents of a circle intersect in the centre of that circle. Thus
in the annexed figure, the straight lines B O, C o drawn at right
angles to the tangents PQ, R S respec-
tively, intersect in the point o, the cen-
tre of the circle A B C. The straight
lines B O, CO are radii of the circle A B C;
and thus, conversely, if a straight line be
drawn through the extremity of a radius
that meets the circumference of any
circle, at right angles to that radius, the
straight line thus drawn at right angles
to the radius is a tangent of that circle.
The ratio of the diameter of a circle to
Р
its circumference is as 1 to 34, or very
nearly so, expressed in figures in the
simplest way possible—that is, the circumference of a circle
is equal in length to three times the length of the diameter,
and one-seventh of its length; or it may be written as 7 to 22,
which means that if a diameter of a circle be divided into 7
parts, the length of the circumference is equal to 22 of those
parts. Expressed in decimals, the ratio is as 1 to 3.14159.
Of course a greater degree of exactness may be obtained by
increasing the number of decimal places in the above, but the
number given is sufficient for all practical purposes.

B

Fig. 52.

The following remarks have been made on the ratio of the diameter of a circle to the circumference by General T. Perronet Thompson, whose name has been already mentioned in our Lessons in Music :

"If it is asked, what after all is the proportion between the diameter of a wheel and its circumference? it is as 1 to 3.14159; etc. etc., to as many figures of decimals as anybody shall think it worth while to discover and add; but, as in the case of the square root of 5, coming to the end, No! As the Irish sailor said of the rope, the end is cut off. It is not quite so easy to add figure to figure as in the case of the square root of 5, but the conclusion is the same. The simplest proportion for common purposes is as 7 to 22. The next, which there is very seldom any occasion to go beyond, is as 113 to 355, on which may be given a useful piece of what is called 'artificial memory.' The ablest man it was ever my chance to know, professor of mathematics in the University of Cambridge, wanted this proportion one day, and was observed to be fidgeting with a pen and a piece of paper. At last he broke out, 'There it is, sir. Write down the first three odd figures in pairs, and cut them in two

113 355.'

I remember telling this to the driver of a French cabriolet on the Pont Neuf, to his great delight. He will never forget it; nor should any working man to whom it may be ever likely to be of use."

READING AND ELOCUTION.-XIV.
ANALYSIS OF THE VOICE (continued).
EXERCISES ON INFLECTIONS.

Rising Inflection.

ference of the circle B C D, which overlaps the circle A C E, cuts Rule 1.—“ High rising inflection."
the circumference of the last-named circle in the points c and F.
When, therefore, two circles touch one another, they touch or

meet in one point only. They cannot touch
each other by any possibility whatever in
more points than one. When one circle is
said to cut another circle, the circumference
of the one cuts the circumference of the
other in two points only; it cannot by any
F
possibility whatever cut the circumference
of the other in more points than two.
Fig. 51.
In Fig. 50, the straight line x Y which
touches the circle A B C in the point B is called a tangent to that
circle. Now in speaking of a line in plane geometry as the
tangent of or to a circle, nothing more is meant than this, that
it touches the circle. The word tangent is derived from the
Latin tango, I touch. In Fig. 52, the straight lines P Q, R s, are
tangents to the circle A B C, P Q touching it in the point B, and

Há!-say you só?

What!-confer a crown on the author of the public calamities ?
Indeed!-acknowledge a tráitor for our sovereign ?

Rule 2.-" Moderate rising inflection."

In every station which Washington was called to fill, he acquitted himself with honour.

As the evening was now far advanced, the party broke up. Where your treasure is, there will your heart be also. Though we cannot discern the reasons which regulate the occur. rence of events, we may rest assured that nothing can happen without the cognisance of Infinite Wisdom.

Despairing of any way of escape from the perils which surrounded him, he abandoned his struggles, and gave himself up to what seemed his inevitable doom.

Had I suffered such enormities to pass unpúnished, I should have deemed myself recreant to every principle of justice and of duty.

Note and Exception.-" Words and phrases of address."

Listen, Amèricans, to the lesson which seems borne to us on the very air we breathe, while we perform these dutiful rights. Ye winds, that wafted the pilgrims to the land of promise, fan, in their children's hearts, the love of freedom! Blood which our fathers shed, cry from the ground;-echoing arches of this renowed hall, whisper back the voices of other days;-glorious Washington! break the long silence of that votive canvas;-speak, speak, marble lips: teach us THE LOVE OF

LIBERTY PROTECTED BY LAW !

Rule 3.-Note.-" Poetic series."

Power, will, sensation, mémory, failed in turn,
Oh! the dread mingling, in that awful hour,
Of all terrific sounds!-the savage tone

Of the wild horn, the cannon's peal, the shower
Of hissing darts, the crash of walls o'erthrown,
The deep, dull, tambour's beat!

All the while,

A ceaseless murmur from the populous town,
Swells o'er these solitudes; a mingled sound
Of jarring wheels, and iron hoofs that clash
Upon the stony ways, and hammer clang,
And creak of engines lifting ponderous bulks,
And calls and cries, and tread of eager feet
Innumerable, hurrying to and frò.

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If guilty, let us calmly abide the results, and peaceably submit to our sentence; but if we are traduced, and really be innocent, tell ministers the truth-tell them they are tyrants; and strain every effort to avert their oppression.

Heaven has imprinted in the mother's face something beyond this world, something which claims kindred with the skies,-the angelic smile, the tender look, the waking, watchful eye, which keeps its fond vigil over her slumbering babe.-In the heart of man lies this lovely picture; it lives in his sympathies; it reigns in his affections; his eye looks round, in vain, for such another object on earth.

Falling Inflection.

Rule 1.-"Intensive downward slide."

Up! all ye who love me! BLOW on BLOW! And lay the outlawed felons LOW! "MACGREGOR! MACGREGOR!" he bitterly cried.

ON! countrymen, ON !-for the day,

The proud day of glory,-is come!
TO ARMS! gallant Frenchmen, to ARMS!

Oh! SHAME on us, countrymen, shame on us ALL!

If we cringe to so dastard a race!

TREMBLE, ye traitors! whose schemes

Are alike by all parties abhorred,—

TREMBLE! for, roused from your parricide dreams,

Ye shall soon meet your fitting reward!

Rule 2.-"Full" falling inflection, in the cadence of a sentence.

Onward still the remote Pawnee and Mandan will beckon, whither the deer are flying, and the wild horse roams, where the buffalo ranges, and the condor soars,-far towards the waves where the stars plunge at midnight, and amid which bloom those ideal scenes for the perse- The changes of the year impart a colour and character to our cuted savage, where white men will murder no more for gold,* nor thoughts and feelings. startle the game upon the sunshine hills.

Rule 4.-"Questions which may be answered by Yes or No." Has not the patronage of peers incréased? Is not the patronage of India now vested in the crown? Are all these innovations to be made to increase the influence of the exécutive power; and is nothing to be done in favour of the pópular part of the constitution, to act as a counterpoise?

Your steps were hasty ;-did you speed for nothing?
Your breath is scanty ;-was it spent for nothing?
Your looks imply concern;-concern for nothing?

Exception." Emphasis."

Tell me not of the honour of belonging to a free country,-I ask, does our liberty bear generous fruits?

Was there a village or a hamlet on Massachusetts Bay, which did not gather its hardy seamen to man the gun-decks of your ships of war? Did they not rally to the battle, as men flock to a feast?

To a lover of nature and of wisdom, the vicissitude of seasons conveys a proof and exhibition of the wise and benevolent contrivance of the Author of all things.

He who can approach the cradle of sleeping innocence without thinking that "of such is the kingdom of heaven," or see the fond parent hang over its beauties, and half retain her breath, lest she should break its slumbers,-without a veneration beyond all common feeling, is to be avoided in every intercourse of life, and is fit only for the shadow of darkness, and the solitude of the desert.

Exception." Modified cadence."

This monument is a plain shaft. It bears no inscription, fronting the rising sun, from which the future antiquarian shall wipe the dust. Nor does the rising sun cause tones of music to issue from its summit. But at the rising of the sun, and at the setting of the sun, in the blaze of noon-day, and beneath the milder effulgence of lunar light, it speaks, it acts, to the full comprehension of every British mind, and the awakening of glowing enthusiasm in every British heart.

Is there a man among you, so lost to his dignity and his duty, as perhaps delight in the perils of martyrdom. I speak not to those to withhold his aid at a moment like this?

Rule 5.-"Penultimate Inflection."

All is doubt, distrúst, and disgrace; and, in this instance, rely on it, that the certain and fatal result will be to make Ireland hate the connection, contemn the councils of England and despise her power.

I am at a loss to reconcile the conduct of men, who, at this moment, rise up as champions of the East India Company's charter; although the incompetence of that company to an adequate discharge of the trust deposited in them, are themes of ridicule and contempt to all the world; and although, in consequence of their mismanagement, connivance, and imbecility, combined with the wickedness of their servants, the very name of an Englishman is detested, even to a proverb, through all A'sia; and the national character is become disgraced

and dishonoured.

It will be the duty of the historian and the sage, in all ages, to omit no occasion of commemorating that illustrious màn; and, till time shall be no more, will a test of the progress which our race made in wisdom and in virtue, be derived from the veneration paid to the immortal name of Washington.

Exception." Emphasis."

Let us bless and hallow our dwellings as the homes of freedom. Let us make them, too, the homes of a nobler freedom,-of freedom

• The penultimate inflection of a concluding series, or of a clause that forms perfect sense, is the same in kind with that which precedes a period, except in verse and poetic prose, which, in long passages of t beauty, retain the suspensive slide.

I speak not to you, sir, of your own outcast condition.-You around me, who, in their persons, their substance, and their families, have endured the torture, poverty, and irremedial dishonour. They may be meek and hallowed men,-willing to endure.

The foundation on which you have built your hopes, may seem to you deep and firm. But the swelling flood, and the howling blast, and the beating rain, will prove it to be but treacherous sand.

Rule 3.-"Moderate" falling inflection, of complete sense. Joy is too brilliant a thing to be confined within our own bosoms: it burnishes all nature, and, with its vivid colouring, gives a kind of factitious life to objects without sense or motion.

When men are wanting, we address the ànimal creation; and, rather than have none to partake our feelings, we find sentiment in the music of birds, the hum of insects, and the lowing of kine; nay, we call

on rocks and streams and forests to witness and share our emotions.
I have done my duty-I stand acquitted to my conscience and
my country-I have opposed this measure throughout; and I now
protest against it, as harsh, oppressive, uncalled for, unjust, -as esta
blishing an infamous precedent, by retaliating crime against crime,—
as tyrannous, cruelly and vindictively tyrannous.

Exception." Plaintive expression."

I see the cloud and the tempest near,
The voice of the troubled tide I hear;
The torrent of sorrow, the sea of grief,
The rushing waves of a wretched life.
No deep-mouthed hound the hunter's haunt betrayed,
No lights upon the shore or waters played;

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