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5. Ganz wie, with a verb, signifies "precisely," or "just as," or "like," as :-Er ist ganz wie ich, he is just as I (am), he is just like me. Sie denkt ganz wie er, she thinks precisely as he (thinks), she thinks precisely like him.

6. Noch, besides meaning "nor," when used in conjunction with weter, "neither," is variously rendered by "still some, or yet more, another, besides," etc., as:-Er schläft noch, he sleeps still. Gieb rem Kinde noch Brod, give the child some more bread. Wann hat er noch ein Pferd gekauft? when did he buy another horse ? Einen Apfel hat das Kind gegessen, aber es hat noch einen, the child has eaten one apple, but it has one besides (or another).

7. Michr, connected with a negative word, is used like its equivalent "more," as :-) -Ich habe keins mehr, I have no more. Ich habe nicht viel mehr, I have not much more. Used with a noun, the adverb follows, while in English it precedes the

noun.

8. Anter signifies other, in the sense of different; it must not be used in phrases like, "I saw him the other day," which is in German, Ich sah ihn neulich (literally, recently); or, Ich sah ihn vor einigen Tagen (literally, a few days ago).

9. The neuter anderes, preceded by etwas (in conversation usually contracted to was), is rendered by the phrase "another thing," as:-Das ist etwas Anderes, or, das ist was Anteres, that is another thing.

10. The adverb anders is readily distinguished by its form, and is rendered by "otherwise, differently," etc., as:- -Er spricht anters, als er denft, he speaks otherwise than he thinks.

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11. Der Staat Pennsylvanien liefert eben so viel Kohlen, als ganz England. 12. Arbeitet Gustav nicht eben so viel, wie sein Bruder Hermann? 13. Die kleine Elise gab ihrer Schwester Pauline eben so viel Pflaumen, wie ihrer Freundin Emma. 14. Haben unsere Nachbarn noch keinen Garten? 15. Nein, sie haben noch keinen. 16. Bleiben Sie noch lange auf dem Lande ? 17. Ich bleibe noch eine kurze Zeit da, und meine Freunde auch. 18. Gehen Sie heute noch spazieren? (Sect. LXIV. 1.) 19. Nein, denn ich muß noch arbeiten. 20. Die Freudenthränen der lang getrennten Freunde rührten tie Herzen aller Zuschauer. 21. Können Sie die Waaren nicht billiger verkaufen? 22. Es ist rein unmöglich. 23. Sie müssen dieses anders machen. 24. Was kann ich anders thun? 25. Du kannst anders reden und handeln. 26. Ich werde Sie besuchen, wenn Sie es erlauben. 27. Er erzählte die Sache ganz anders. 28. Es ist etwas anderes, ob ich schreibe: er ist gelehrt," oder „geleert.“

EXERCISE 59.

1. Has the teacher taken away the paper or the book? 2. He has taken away both; for both belong to him. 3. Both towns are situated on navigable rivers. 4. They may take either way, as they have proceeded so far. 5. A great part of the land in America is still uncultivated. 6. He who wants the purpose, must will the means. 7. The Rhine steamboat has just started for Holland. 8. You err altogether when you say that you have quite surmounted every difficulty, otherwise all that you have stated would be correct. 9. Which of us is right, I or he? 10. You are both wrong. 11. It is quite another thing to say that he was not well, and could not come in consequence of it. 12. I shall speak no more about it; because I have found upon closer investigation, that he is neither covetous nor prodigal. 13. They do not think themselves better than others. 14. to Emma is just as intelligent as Eliza. 15. The sailor sets sail for America to-morrow. 16. Do you drink wine or beer? 17. I drink neither wine nor beer, I always drink water. 18. Gustavus gave the boy a thaler to buy some coals for his mother. 19. Pennsylvania is a rich and flourishing state in the United States of America. 20. She is just like her sister. 21. Give the boy some more plums. 22. I have no more. 23. The girl shed tears of joy when she saw her mother. 24. That ware is cheap, and the pattern of it is beautiful. 25. My friend has purchased a new winter coat. 26. This merchant sends his goods to the town in a wagon. 27. Will you take a walk to-morrow? 28. It is impossible.

take a walk.
Staat, m. state.
Thaler, m. thaler (a
German
coin,
worth about 3s.)
Trennen, to separate.
Un'angebaut, unculti-
vated.

Unmöglich, impossible
Unrecht, wrong.
Verständig, intelli-

gent, sensible.
Waare, f. ware, goods.
Weg'nehmen, to take

away.

Win'terrock, m. winter
coat.

Sache, f. thing, af- Wohlfeil, cheap.
fair.
Zuschauer, m. specta-
Som'merrock, m. sum- tor.
mer coat.
RÉSUMÉ OF EXAMPLES.

Er hat zwei Söhne, aber beite sind taubftumm.

Der Riese faßte die Keule mit bei
den Händen.

Hat der Kaufmann ein Pferd over
einen Wagen?
Er hat Beides.

Die Wahrheit und die Rose sind
sehr schön, aber Beide haben Dor-

nen.

Ein aufrichtiger Mann verab'scheut eine Lüge.

dant jeter Mensch hat e'ben so viel Kummer als Freude.

3wed,m.aim,purpose.

He has two sons, but both are
deaf and dumb.

The giant seized the club with
both hands.

Has this merchant a horse or a
wagon ?

He has both.

The truth and the rose are
very beautiful, but both have
thorns.

An upright man abhors a lie.

Nearly every human being has
quite as much sorrow as joy.

EXERCISE 58.

1. Wollen Sie ein Muster von diesem oder jenem Tuche haben? 2. en keinen von beiden. 3. Wir geben ihm einen Thaler für jeden der beten Männer. 4. Trinken Sie Wein oder Bier? 5. Ich trinke weber Wein noch Bier (or, ich trinke keines von beiden). 6. Sie haben Recht, daß Sie das gethan haben. 7. Ist es recht, daß Johann so lange ausbleibt ? 8. Nein, es ist unrecht von ihm, da er seine Aufgaben zu lernen bat. 9. Wie viel Tuch braucht der kleine Friedrich zu (Sect. LXXIII. 1) einem Sommerrocke? 10. Er braucht eben so viel, wie zu einem Winterrocke.

MECHANICS.-VIII.

THE THREE ORDERS OF LEVERS.-THE COMMON
BALANCE.

OF Levers there are commonly reckoned three kinds, of which
Figs. 45, 47, and 49 furnish illustrations, in which the bar
extends to equal distances on either side of the fulcrum, F;
in order that, the centre of gravity being supported at F, it
may not by its weight interfere with the action of the Weight
and Power. In that case
we must consider the true
lever only as so much of the
bar as is between P and w,
or P or w and F, the points
of application and the ful-
These levers are said to be of three orders.
First Order.-When the fulcrum is between the Power and
Resisting Weight.

crum.

F

P

W

Fig. 45.

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First Order.-The Condition of Equilibrium in this we have already determined in connection with the balls in Lesson Fig. 46. VI. (Fig. 44). The Resisting Weight (Fig. 45) is to the Power inversely as w F to P F, or the weight multiplied by the arm, w F, is equal to the power multiplied by the arm P F. Of this kind of lever the examples are very numerous. In Fig. 46 the crowbar is used as a lever, by means of the fulcrum in the middle, to lift the chest, the push of the hand and the weight of the chest, both pa

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LESSONS IN GERM

SECTION XXXL-INSEPAR

BESIDES the separable particles (Se
class (be, emp, ent, er, miß, ver, etc.,
are never used apart from the ra
prefixed, and hence are called inse
union of these particles be, emp,
fehlen, etc., we have the compon
erholen, mißfallen, verhören, zermalm
tion to the English compound
take, etc. With few exception
German, unlike most English r
as in combination with prefi
to demolish.

Many particles in German verbs, have their exact equ interpret; mißreuten, to mi befall, etc. (§ 97. 1, 2, et In German, as in Englis the primary accent. (§

1. Bor, which is often the latter, always preced as:-Er war vor zwei (literally, he was here Seit (since), when answers to "for" or he has been (literall Ich habe ihn seit einem during a whole year

Ant'worten, to answ (intransitive). Beantworten, to

swer (transiti

Begrün'ten, to co

tute.

Beschreiben, to scribe. Betra'gen, to be Erfin'den, to in Erhalten, to r Erniedrigen, t

Diesen schöne mir mein gege'ben.

Die Freunde

ten bege'b

Die feindlich
ben.
Der Lehrer
ben.

1. T

er hat

Kna

get

(

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ventum.jr baksom
orazitage ponać by

that the earths of the
sther numbers in the serial
& the serund, 5 to 2 in the 1

zemnú, at › support at w? It is not

ITO khơm Komething about multiplying?
the first amor by the principle of moments,
the pond at Pas equal to 4 times the

tat prand in the second arver at P, ; that w
wer of a pound. But this force, fot
mplied at r, in the proportion of t

of of a pound, and this evans att at w off of off of that unit. ton, the 1 pound balances 13 pounds marse, what i: true of these numbers is tml and the rule you arrive at is this—

a for ratios of the Power arms to the Resistance arUS, MUT E-Multiply together the fractions which 1 obtained is the number of pounds of the Resistang t pound of the Power balances. When the Powe s 1 pound, multiply this number into that of the p fractions of a pound in it.

And this leads us to another result, which expresse tion between the power and resistance without_practi in the above example, we had the resistance equ of of the power, it is evident that the three den multiplied into the resistance must be equal to the thr rators into the power, and thus, extending the pract may say that

The Power multiplied by the several imgths of the arms is equal to the Resistance maphed by those of the

ance arms.

for a single lever. And observe that this ta And you thus have a result net ulike tror establebe. only for a combination of keress of the v good of other combmations mar or all of the third, or of the i

D

G

le for each kind, and therefore → reason I have above avoided, rinciple, the terms "long arm cead "power arm" and "resistthe arms that work with the

n of levers which is most likely aon weighing machine, used for 3, or luggage at railway stations. In Fig. 52 is a ground-plan of this piece of mechanism, where at A, B, C, D, the four corners of the bottom of a shallow box, are the fulcrums of four levers of the second order, which meet, two and two, on either side at F, and are joined across by a stout steel pin, by which they are also connected with the lever of the second order, E G, which has its fulcrum at E. The end, G, of this lever is connected by a rod which ascends perpendid, and is attached above to the short one of the first order, generally a steelescribed to the longer arm of which the is attached. We thus have a triple comfirst four at the bottom, by being united ne lever. On these four at a, b, c, d, are d steel, presented upwards, on which rests atform, on which the cart or luggage to be The weight pressing at a, b, c, d, tends to end, F, of the four levers, and with it also rer E F G. The latter tries to pull down the e short arm of the steelyard above, which the counterpoise on the longer arm of the g equilibrium, and making known the weight age.

king the four platform-levers as one, suppose as in the combination are each one-fifth of the evidently, as proved above, the resistance is etimes into the power-that is to say, 1 pound teelyard balances 125 pounds, or 1 cwt. and 13 latform. If the proportion were one-eighth, it cwt. 8 pounds, which strikingly illustrates the antage gained in these machines. We will now ommon balance, and, in the next lesson, examine of other weighing instruments, bent levers, and axle, and their combinations.

THE COMMON BALANCE.

ng instruments, the scale, or common balance, first attention. It is a lever of the first order, in counterpoise, or power, is equal to the resistance, or weighed. There is first the beam, A B, at the ends of 53) are the hooks, from which hang the chains or ch support the pans or scales below. Since the n the scales are required to be equal, the fulcrum, F, e in the middle of the beam, equally distant from the suspension of the chains, else the balance is frauduthe purchaser who has his tea or sugar served to him e end of the longer arm is getting less than his money's I shall direct your attention to the case in which the ining the points, A B, of suspension passes through the ting point of the fulcrum, as it is the simplest; and des of this kind, as you will see, have a peculiar advantage their sensibility.

ww, it is evident, since A B is bisected at F, and the scales, as, and weights on either side are equal forces, that whatbe the position in which I place the beam, the resultant of e forces must pass through F, and, being there resisted, re the whole apparatus at rest. Moreover, if the centre of wity of the beam is at F, so far as its weight is concerned, tre will be equilibrium in every position. But such a pair scales would be utterly useless, since, for equal weights, the ns should rest only in an horizontal position. How, then, is this latter object accomplished? By having centre of gravity of the beam below the fulcrum, when the

arms are horizontal. The desired position is then one of stable equilibrium (see Lesson VII.), to which the beam will revert when displaced from it, and in which the line F G is perpendicular to the line AB, joining the points of suspension of the scales. For a good pair of scales, therefore, there must be stability as well as accuracy.

But a balance should also be sensitive-should indicate a slight difference of weights in the scales. How is this secured? Suppose the scales equally loaded, and that a slight additional weight (call it P), is thrown into the scale a in Fig. 53, causing it to decline through some angle agreed upon as sufficient to indicate a difference of weights to the eye. As a descends, the centre of gravity, G, of the beam ascends at the other side, until its weight (call it w), acting at G, balances P. We have thus a new lever, A D, the fulcrum of which also is F, and at whose ends the forces P and w act. And since in that case, as proved in the last lesson, P multiplied by A F must be equal to w multiplied by F D, the length A B, and the weight w, of the beam being the same in any number of balances in a manufactory, that one which moves through the angle agreed on, with the smaller additional weight P, must also have F D smaller; or, which comes to the same thing, since the angles of the triangle FG D are given, that at F being a right angle, it must have F G smaller. Everything else, therefore, being the same, that balance has the greater sensibility, the centre of gravity of whose beam is as little as possible below the fulcrum. Summing up, then, we have for the requisites of a good balance the following:1. For Accuracy.-That the arms be equal.

2. For Stability and Horizontality. That the centre of gravity of the unloaded beam be below the fulcrum, on a line through its supporting point, perpendicular to that which joins the points of suspension of the scales.

3. For Sensibility. That the centre of gravity of the beam be as little as possible below the fulcrum.

You will observe that the second and third conditions oppose each other. The lower the centre of gravity is below the fulcrum, the greater is its stability, but the less its sensibility. Both qualities are essential, and are therefore secured only by a compromise; the centre for sensibility may approach the fulcrum, but not too close; for stability it keeps off, but not too far.

Further, observe the consequence of making the line joining the points, A B, of suspension pass through the fulcrum. However the pans are loaded, it is only the difference (P) of the weights in them that affects the sensibility. The resultant of the lesser one in B, and of as much of that in A as is equal to it, passes through and is resisted by F, and affects neither stability nor sensibility. If A B were not to pass through F, then these weights would have influence as regards these qualities, but that kind of balance we are not here considering.

A most important question is, how to detect fraud

Fig. 53.

in a pair of common scales. The arms in that case not being equal, all the purchaser has to do, if he doubts the honesty of his tradesman, is, after the first weighing, to make the shop weight and the substance weighed change pans. If the two balance each other equally as before, the scales are honestthe arms are equal; but if not, fraud is proved.

But how, in that case, may the purchaser still get his true pound of tea, or sugar, or other commodity? The shop weight being supposed true, the imperial stamped weight, let the deficient tea be weighed as before from the longer dishonest arm. Leaving it then in the scale, let him require the shopman to remove the weight from the other scale, and fill it with tea until that in the first ne is balanced. He now has a true pound of tea balancing the deficient pound, as the imperial weight first did. Let him carry off this pound, and he has his money's worth.

But there is another way by which the purchaser may not

only get his due quantity, but turn the tables on the vendor, and by the very fraudulent balance itself get more than his money's worth. Suppose he is buying two pounds; then let him have one pound weighed in one scale and the other pound in the other scale; it so happens that invariably the two together are more than two pounds. The reason you will understand by an example. Suppose one arm is 14 inches long, and the other 15 inches. Then, weighed at the latter arm, the purchaser gets only ths, which is less, but at the former ths, which is more than one pound. But by the latter he gains ath of a pound more than he is entitled to, while at the former he loses only th. So on the whole, since ath is greater than ath, he is a gainer; he has caught the vendor in his own trap. Or, you may add up the two fractions ths and ths, and the sum you will find to be greater than 2 by the fraction And what is true of these numbers is true of all others, which represent the proportion of the arms-what you lose at the long arm is more than recompensed by what you gain at the short one.

READING AND

ELOCUTION.-IX.

ANALYSIS OF THE VOICE (continued).

II. DUE QUANTITY, OR LOUDNESS.

THE second characteristic of good reading, is the use of that degree of loudness, force, "volume," or "quantity," of voice which enables those to whom we read or speak, to hear, without effort, every sound of the voice; and which, at the same time, gives that degree of force which is best adapted to the utterance of the sentiments which are read or spoken.

All undue loudness is a great annoyance to the ear, and an injury to the expression; while a feeble and imperfect utterance fails of the main purposes of speech, by being partly or entirely inaudible, and consequently utterly unimpressive.

The failure, as regards loudness, is usually made on passages of moderate force, which do not furnish an inspiring impulse of emotion, and which depend on the exercise of judgment and discrimination, rather than of feeling.

It is of great service, however, to progress in elocution, to possess the power of discriminating the various degrees of force which the utterance of sentiment requires. The extremes of very "loud" and very "soft," required by peculiar emotions, have been exemplified in the exercise on "versatility" of

voice.

There are three degrees of loudness, all of great importance to the appropriate utterance of thought and feeling, required in the usual forms of composition. These are the following "6 Moderate," "forcible," and "impassioned." The first, the "moderate," occurs in the reading of plain narrative, descriptive, or didactic composition, addressed to the understanding, rather than to the feelings; the second, the "forcible," is exemplified in energetic declamation; the third, the "impassioned," occurs in the language of intense emotion, whether in the form of poetry or of prose.

Watchful attention will be required, on the part of the student, in practising the following examples, so as to enable him to detect, and fix definitely in his ear, the exact degree of loudness appropriate to each passage. The exercises should be repeated till they can be executed with perfect precision, so as to form a standard for all similar expression, in subsequent reading.

Exercise in "Moderate" Force.

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happiness of kindred, and parents, and children. We welcome you to
the immeasurable blessings of rational existence, the immortal hope of
Christianity, and the light of everlasting truth!
Impassioned" Force.

Shame shame! that in such a proud moment of life,
Worth ages of history,-when, had you but hurled
One bolt at your bloody invader, that strife

Between freemen and tyrants had spread through the world,

That then,-oh! disgrace upon manhood!-e'en then
You should falter,--should cling to your pitiful breath,-
Cower down into beasts, when you might have stood men,
And prefer a slave's life to a glorious death!

It is strange!-it is dreadful!-Shout, Tyranny, shout,
Through your dungeons and palaces, "Freedom is o'er !"-
If there lingers one spark of her fire, tread it out,
And return to your empire of darkness once more.

III.-DISTINCT ARTICULATION.

Correct articulation is the most important exercise of the voice and of the organs of speech. A reader or speaker, possessed of only a moderate voice, if he articulates correctly, will be better understood, and heard with greater pleasure, than one who vociferates. The voice of the latter may, indeed, extend to a considerable distance; but the sound is dissipated in confusion of the voice of the former not the smallest vibration is wasted-every sound is perceived at the utmost distance to which it reaches; and hence it even penetrates farther than one which is loud, but badly articulated.

In just articulation, the words are not hurried over, nor precipitated syllable over syllable; nor, as it were, melted together into a mass of confusion; they are neither abridged nor prolonged; nor swallowed, nor forced, and, if I may so express myself, shot from the mouth; they are not trailed nor drawled, nor let slip out carelessly, so as to drop unfinished. They are delivered out from the lips, as beautiful coins newly issued from the mint, deeply and accurately impressed, perfectly finished, neatly struck by the proper organs, distinct, sharp, in due succession, and of due weight.

This department of correct reading belongs, properly, to the stage of elementary lessons. But negligence in general habit, and remissness in early practice, are extensively the causes of an imperfect articulation.

A paragraph or two of every reading lesson should, previous to the regular exercise, be read backward, for the purpose of arresting the attention, and securing every sound in every word.

The design of the present lesson does not admit of detail in the department of elocution now under consideration. The importance, however, of a perfectly distinct enunciation can never be impressed too deeply on the mind of the student. An exact articulation is more conducive than any degree of loudness to facility of hearing and understanding. Young readers should be accustomed to pronounce every word, every syllable, and every letter, with accuracy, although without laboured effort. The faults of skipping, slighting, mumbling, swallow. ing, or drawling the sounds of vowels or of consonants, are not only offensive to the ear, but subversive of meaning, as may be perceived in the practice of several of the following examples.

Examples.

1. That lasts till night: that last still night.

2. He can debate on either side of the question: he can debate on neither side of the question.

3. The steadfast stranger in the forests strayed.

4. Who ever imagined such an ocean to exist ?-Who ever imagined such a notion to exist?

5. His cry moved me: his crime moved me.

6. He could pay nobody: he could pain nobody. 7. Up the high hill he heaves a huge round stone. 8. Tho' oft the ear the open vowels tire.

9. Heaven's first star alike ye see.

The following description of a whale chase, taken from Goodsir's "Arctic Voyage," will furnish a useful exercise in distinctness of articulation. Read it with animation and "moderate force," but not too fast.

We pulled in the direction in which the whale was "heading," where the rest of the boats already were; before we got up to them, she had made her appearance at the surface; a second boat had got fast to her,

She

and just in time, as she was seen to be "loose" from the first. did not take out much line from this boat, but remained away a considerably longer time than usual, greatly to our astonishment, until we found that she was "blowing" in some holes in the floe, a good distance from the edge of it. One of the harpooners immediately proceeded over the ice with a hand-harpoon, trailing the end of the line with him, assisted by part of his crew, and from the edge of the hole drove his weapon into the body of the poor whale; whilst some of the others following plied the bleeding wretch with their long lances, so that she was soon obliged to betake herself again to the open water outside the floe. Here more of her enemies were waiting, for one boat was immediately upon her, and a gun-harpoon was at once driven almost out of sight into her huge side, which was already bristling with weapons. Our boat was on her very back as she dived with an unwieldy roll, which sent it surging gunwale under, taking the line whistling out for a score of fathoms, until the harpooner, knowing she was pretty well exhausted, stopped her way, by taking three or four turns round the "boltard." But every few seconds she would make a start, drawing the boat almost head under, until the line was permitted to run out again, which, as it did so, made a grinding, burring noise, eating deep into the hard lignum vitæ of the boltard, enveloping the harpooner in smoke, and causing the most distinct smell of burning, which was only prevented from actually taking place by the linemanager throwing water constantly on it.

Again she appeared at the surface, but far exhausted; still she made a strong fight for it, lashing about with her tail and fins in fury whenever she seemed to have regained breath. It was no very pleasant sight to see her tail quivering high up in the air, within but a short distance of us, and coming down on the water with a loud, sharp crack, like the report of a dozen rifles, and which, had it alighted on any of our boats, had power sufficient to have converted their timbers into something very like lucifer matches. A few more lances soon settled her; and ere long, she was rolling on her back. The usual cheers of triumph were given, and we had time to breathe and shake ourselves, for it may be believed we had not escaped the showers of spray which the defunct had sent about so liberally. The water far around us was dyed with blood, and covered with a thick pellicle of cil, upon which the Mollys were as busy as they could be, whilst the edges of the ice, as far as we could see, were deeply crimsoned; and a hummock, on the edge of the floe, beside which the final struggle had taken place, was from the summit downward streaked with the black Mood which the last few blasts of the dying monster had sent over it. IV. CORRECT PRONUNCIATION.

That pronunciation is correct which is sanctioned by good usage or custom. Good usage implies the habit of persons of good education, as regulated by the decisions of learning and taste, exemplified in standard dictionaries-a style which is equally free from the errors of uneducated or negligent custom, and the caprices of pedantry-which falls in with the current of cultivated mind, and does not deviate into peculiarities, on the mere authority of individuals. Good taste in pronunciation, while it allows perfect freedom of choice as to the mode of pronouncing words liable to variation in sound or accent, requires a compliance with every fixed point of sanctioned usage.

The subject of pronunciation like the preceding onearticulation-belongs properly to the department of elementary instruction. But as this branch of elocution does not always receive its due share of seasonable attention, many errors in pronunciation are apt to occur in the exercise of reading, as performed by even the advanced classes in schools. To avoid such errors, it will be found useful to discuss, closely and minutely, the correct pronunciation of every word which in any lesson is liable to be mispronounced, the standard of reference being any good dictionary of the English language.

LESSONS IN GEOMETRY.—IX. IN the construction of triangles the student has learnt, by Problem XVI. (page 209), how to draw an equilateral triangle of any dimensions, the only two data (or facts given from which other facts may be deduced) that are required in the formation or construction of an equilateral triangle being, the length of one of its three equal sides on the one hand, or its altitude on the other.

It will be remembered that, in Definition 18 (page 53), it was stated that triangles are classified according to the relation of their sides, as

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Now, as the three interior angles of a triangle are together equal to two right angles or 180 degrees, and as an obtuse angle is any angle greater than a right angle or 90 degrees, while an acute angle is any angle less than a right angle or 90 degrees, it is manifest that-

An equilateral triangle must necessarily be an acute-angled triangle, since it has three equal angles, each of which is less than 90 degrees, being one-third of 180 degrees; while

An isosceles, or a scalene triangle, may be a right-angled triangle, or an obtuse-angled triangle, or an acute-angled triangle.

To proceed still further into an analysis of the conditions under which the different kinds of triangles will appear, it may be said that—

I. An acute-angled triangle may have

1. Three sides equal, and three angles equal, when it is an equilateral triangle.

2. Two sides equal, and two angles equal, when it is an acuteangled isosceles triangle.

3. All its sides unequal, and all its angles unequal, when it is an acute-angled scalene triangle.

II. An obtuse-angled triangle may have

1. Two sides equal, and two angles equal, when it is an obtuseangled isosceles triangle.

2. All its sides unequal, and all its angles unequal, when it is an obtuse-angled scalene triangle.

III. A right-angled triangle may have

1. Two sides equal, and two angles equal, when it is a right-angled isosceles triangle.

2. All its sides unequal, and all its angles unequal, when it is a right-angled scalene triangle.

We have already learnt, as it has been said above, how to draw an equilateral triangle of any dimensions, the conditions necessary for its construction being given. Let us now see what data we require to enable us to draw any isosceles or scalene triangle characterised by having one right angle, one obtuse angle, or three acute angles.

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To determine any isosceles triangle, it is plain that we must have one or the other of the following series of data. I. With regard to the sides without the angles :1. The length of the two equal sides, and the length of the third side or base.

2. The length of the two equal sides, and the altitude of the triangle. 3. The length of the base, and the altitude of the triangle.

II. With regard to the sides and angles combined :

4. The angle at the vertex of the triangle, and the length of the two equal sides.

5. The angle at the vertex of the triangle, and the length of the base.

6. The angle at the vertex of the triangle, and the altitude. 7. The equal angles at the base, and the length of the equal sides. 8. The equal angles at the base, and the length of the base itself. 9. The equal angles at the base, and the altitude.

In any case, when the length of the sides or altitude is given, either with or without the extent of the opening of all or any of its angles, an isosceles triangle can be constructed, which is the only form of the isosceles triangle which will satisfy the particular requirements laid down in the data; but where the angles only are given, an endless number of triangles similar in form,

but of different superficial areas, may be drawn, all of which shall satisfy the general requirements set forth in the data, for it must be remembered that the size of an angle is determined by the extent of the opening between the lines that form its sides, and not by the length of its sides; and this leads us to the construction of an isosceles triangle under general conditions, namely

:

III. With regard to the angles without the sides :-10. The angle at the vertex of the triangle.

11. The equal angles at the base of the triangle.

The first case named above of the construction of the isosceles

triangle, when the length of the two equal sides, and that of the third side is given, is met by Problem VIII. (page 191)

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