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the shape of a cone, the apex of which is truncated; next cut
a notch in the lower or base edge of the cone, and the stand is The words printed in italics form an adverbial phrase. Adverbial
In great concern he ran to bear the sad tidings.

made. The use of it will be evident from an examination of

the diagram, fig. 28. The notch admits the gas delivering
tube, the truncated apex delivers the gas into the bottle, which
rests supported on the sides.

If the student were not told of these contrivances he might
think me remiss; but I want to create a feeling of independence
in his mind, to impress him with the conviction, that in
the majority of chemical operations involving the use of
mechanical contrivances, many different methods admit of
being followed, each equally good. The support just described
is useful, and not inelegant, but I shall not quarrel with a
student who tells me that two bricks set edgeways in a pan of
water, fig. 29, furnish a support which is nearly as good.

Fig. 29.

phrases involve what may be called an adverbial object; thus, in great concern is an adverbial object. Adverbial objects may be

various; as,

1. Of time:
2. Of place:

On arriving home I hastened to bed.
He slew his foe in the dell.

3. Of manner: The father begged his life with many supplications.
An object, then, may be not only single or compound,
Single: He launched the ship;

Compound: The waves overwhelmed the boat and the crew; near or remote; as,

Near:

He sold his desk;

Remote: He sold his desk to his clerk;

but also adverbial, and that of three kinds,-of time, of place, of

manner.

A simple sentence is a sentence which has one subject and one affirmation or predicate; and a compound sentence is a sentence that has more than one subject and more than one predicate. The component parts of a compound sentence are called its members. These members may be two or more; they may also each form a separate sentence:

1

Compound Sentences of two Members.
2

He will perish who loves unrighteousness.

1

2

The lark sang his matins and sank into his nest.

The great fault of most books which treat of chemical ma-
nipulations is this:-they represent the apparatus which is
not intrinsically best for gaining any particular result, but the
apparatus which makes the prettiest engraving. This, in my The first sentence is equivalent to these two propositions :—
opinion, is but a questionable benefit to the pictorial art, and
a vast disadvantage to the student of chemistry.

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1. Some one will perish.

2. The lover of unrighteousness will perish.

The second sentence is equivalent to these two statements :

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When the Queen arrived, the fleet had weighed anchor and sailed.
1. The Queen arrived.

2. Before then the fleet had weighed anchor.
3. Before then the fleet had sailed.

a Thus what in the compound sentence stands as three members,
becomes in the analysis three individual sentences.

Predicate. conquered Burmah. subject, which may be

gained respect.

being Queen of England
when Queen of England
on assuming the sceptre
while Queen of England
These accessaries are denominated subject accessaries, because
they qualify the subject. Accessaries may qualify the object
also; e. g.,

Object-Accessaries.

by her virtues.

Victoria gained respect for the good laws she sanctioned.
in consequence of her birth.

It is easy to see that the members may be increased almost at pleasure :

The sick and all but dying man drinks water and revives. Compound sentences have members of two kinds, the principal and the accessary. The principal member is that which enunciates the leading thought, the accessary member is that which enunciates the subordinate thought:

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These accessaries, whether they attach to the subject or the Rel. Pron.: The man object, may be characterised as

Adverbial Accessaries.

ACCESSARY INTERPOSED. who drinks when he drinks if hedrinks

Principal.

is refreshed

is refreshed

is refreshed

Rel. Ad.: The man
Conjunc.: The man
Appended members are added by means of conjunctions,

The essential quality of the adverb is to declare the quality of adverbs, and pronouns:— an affirmation, thus:

He writes well.

ACCESSARY
APPENDED.

Principal.

and is refreshed.

The man is refreshed

But the quality of an act may be assigned by an adverbial phrase Conjunc.. The man drinks
as well as by a simple adverb; e. g,,

Adv.:

when he drinks,

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In number one, who is of the first person, because I is of the first person; who is of the singular number, because I is of the singular number. The effect of the relative on the verb is more clearly seen in the second instance, where an s is added to the verb, which accordingly appears as reflects.

As the language is now written and spoken by the best authorities, the relative who has one change of form in the nominative, namely, in which; which is commonly applied to things. Who, however, has a genitive and an objective, as well as a nominative case, and may be declined or inflected thus:

Singular and Plural.

SINGULAR

AND

PLURAL.

WHO DECLINED.

"The malcontents made such demands as none but a tyrant could refuse."-Bolinbroke.

What is a relative which performs the double function of a subject and an object, being equivalent to that which, and used in only the neuter gender; e. g.,

"My master wotteth not what is with me.”—(Gen. xxxix. 8.)

As a subject for exemplifying the doctrines laid down in regard to the structure of sentences, I shall take some sentences from Daniel Defoe, a writer of idiomatic English.

Compound Sentence.

"Oxford makes by much the best outward appearance of any city I have seen, being visible for several miles round on all sides in a most delightful plain; and adorned with the steeples of the several colleges and churches, which make a glorious show."

Here I must premise that the form "the best outward appearance of any city," &c., is incorrect, and should have been "the best outward appearance of all the cities I," &c. This compound sentence may be reduced into these simple sentences :—

1. Oxford makes a very good appearance.

2. Oxford makes an appearance better than many cities.

3. I have never seen a city with a better appearance than Oxford.

4. Oxford is visible for several miles round.

5. Oxford is visible from all sides.

6. Oxford stands in a most delightful plain.

7. Oxford is adorned with the steeples of several colleges.

8. Oxford is adorned with the steeples of several churches. 9. The architectural decorations of Oxford make a glorious show. The resolution of this long sentence into the several distinct propositions which it contains, has, by showing the meaning of the tions which those parts sustain to each other, thus :— several parts, prepared the way for our exhibiting the logical rela

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a verb in the

Number twelve presents an appended relative accessary sentence of which these are the components; namely, which, a relative pronoun agreeing with its antecedent steeples; make, indicative mood, third person, plural number, agreeing with its subject which; a, the indefinite article limiting show; glorious, an whose of which (whose) adjective qualifying show; show, a common noun dependent on or the object to the verb make. Viewed structurally, this

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Feminine. who

whom

Instead of whose and which we sometimes find whereof.

Neuter which

which

That, which is without any inflexional change, may be used in lieu of who or which, being applied to both persons and things; e. g.,

"He that reproacheth a scorner, getteth to himself shame."(Prov. ix 7.

The word as is also used with the force of a relative after such, many, the same; c. g.,

appendage stands thus:

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the several components. In other terms, convert each of these compound sentences into simple sentences. Distribute each simple sentence into subject and predicate, distinguishing the verb (the copula) and the attribute. Next, exhibit each compound sentence in its several members, showing what are principal, what accessary, and what appended, what interposed; together with the accessaries to the subjects and objects, and the adverbial objects. Finally, give the grammatical analysis of the whole.

ON PHYSICS OR NATURAL PHILOSOPHY. No. VI.

LAWS OF GRAVITY; PENDULUM.

(Continued from page 63.)

Formula relating to Falling Bodies.-The second and third laws of falling bodies may be respectively represented by the formulæ v=gt, and sgt2. For, let g be the velocity acquired at the end of a second by a body falling in a vacuum, and its velocity after t seconds; then, the velocities being proportional to the times, we have g:v: 1:t; whence v=gt (1). Again, a body which falls during t seconds by a motion uniformly accelerated, with an initial velocity equal to zero or 0, and a final velocity equal to gt, will describe the same space as if it fell during the whole time t by a uniform motion, with a mean velocity between O and gt, that is, with the velocity gt. Now, in the latter case, the motion being uniform, the space described is equal to the product of the velocity and the time; whence, denoting this space by s, we have s=}gt ×t=} gt2 (2). The demonstration of these theorems is given mathematically in treatises on Dynamics; sce Whewell's Mechanical Euclid, and other elementary works of the same description.

If in the formula (2) we make t=1, we have sg, whence g2s; that is, the velocity acquired at the end of a unit of time is double the space described in that unit of time. This value of g is called the measure of gravity. Thus, in the latitude of London, it has been found that a body falling near the surface of the earth, in a vacuum, describes about 16 feet in the first second of its fall; hence, the measure of gravity of London is about 324 feet; in other words, after a body has fallen 16 feet in 1 second, by the force of gravity, it would, if the attraction of the earth were removed or counteracted, continue to fall ever after with a uniform velocity of 32 feet per second.

In formula (1) the velocity v is expressed in a function of the time; that is, an expression involving the number denoting the time; but we can likewise express it in a function of the space described, by eliminating t from the two formula (1) and (2). For, from the first, we have t, whence t ; now substi

v

9

v2

g2

tuting this value of in formula (2), we have s9x

v2

2g

; and multiplying both sides of this equation by 2g, we have 2gs; and extracting the root, we have finally, "=√2gs; hence, we conclude that, when a body falls in a vacuum, the velocity acquired at any given instant is proportional to the square root of the height of the fall.

The formulæ v=gt, and s= } gt2, having been determined by considering gravity as an accelerating force, and consequently in a case where motion is uniformly accelerated, they may be considered as general formula for this kind of motion. But it must be observed, that as g denotes the acceleration of the velocity imparted in each second by the accelerating force, the value of g will vary with the intensity of the force.

Causes which Modify the Intensity of Gravity.-Three causes have an effect in making the intensity of gravity vary; 1st, the elevation of the place above the ground, or general level of the earth's surface; 2nd, the centrifugal force due to the earth's rotation on her axis; 3rd, the depression of the earth's surface near the poles.

1. Since terrestrial attraction acts upon bodies as if the whole mass of the globe were collected at its centre, and this attraction acts upon them in the inverse ratio of the square of their distance

from that centre, it follows that the intensity of gravity will increase or decrease, according as the bodies approach to, or recede from, the general level of the earth's surface. This variation, however, is not apparent in the ordinary phenomena which are observed at the surface of the globe, because, its radius being nearly 4,000 miles, the distance from the centre is sensibly the same when a body is elevated by a few hundred yards. But when the heights of bodies above the earth's surface are very considerable, gravity can no longer be considered as having the same intensity. It is necessary, therefore, to remember that the laws of falling bodies already explained are only true for heights within certain appreciable limits.

2. The second cause which modifies the intensity of gravity is the centrifugal force. A force which produces a curvilinear motion, and which gives to bodies under the influence of this motion a tendency to fly off from the axis of rotation, is called centrifugal. It is demonstrated in treatises on Rational Mechanics, that the centrifugal force is proportional to the square of the velocity of rotation; whence it follows that, under the same meridian, it increases as we approach the equator, where it reaches its maximum, because there the greatest velocity takes place. At the poles the centrifugal force is zero. At the equator, the centrifugal force is directly opposed to gravity, and is equal to of its intensity. Now 289 being the square of 17, it follows that, if the motion of rotation in the earth were 17 times slower than it is, the centrifugal force at the equator would be equal to that of gravity, and all bodies on its surface in this latitude would be on the point of being projected into space.

As we proceed from the equator towards the poles, gravity is less and less affected by the centrifugal force. This happens chiefly because the centrifugal force decreases in proportion as we recede from the equator, and also because that, at the equator, the centrifugal force is directly opposite to that of gravity, whereas, in proceeding towards the poles, its direction becomes more and more inclined to that of gravity, and thus loses intensity. Thus, in fig. 15, in which PQ represents the axis of the earth, and EF the Fig. 15.

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3. The intensity of gravity is also modified by the depression of the earth at the poles; for, in the vicinity, and at these points, bodies are nearer to the centre of the earth, and consequently more subject to its attraction.

Measure of the Intensity of Gravity.-After the preceding considerations, gravity may be considered in the same place, and in cases where the heights of the fall are inconsiderable, as a constantly accelerating force; and that the measure of its intensity is the velocity imparted in one second of its fall to a body falling in a vacuum, without regard to its mass, seeing that in a vacuum all bodies fall in the same time. This velocity is represented in general by 2g: it increases from the equator to the pole, and at London it is 32 feet.

The Pendulum.-The general name of pendulum is given to every solid body suspended at one point on a horizontal axis, around which it oscillates. There are two kinds of pendulum; the simple and the compound.

The simple pendulum (which exists only in idea) is that which would be formed by a heavy material point suspended by a per

fectly rigid rod, inextensible and without weight, at a point round which it freely oscillates. Of course this pendulum cannot be put in actual practice, because it is purely theoretical, and is employed only to determine by calculation the laws of the oscillations of the pendulum.

The compound pendulum may be varied in its form in any manner whatever, but it is generally made of a metallic lens or bob, suspended by an iron or wooden rod, and moveable round a horizontal axis, such as the pendulum of a clock, the pendulum , in fig. 13 of the preceding lesson, or that exhibited in the following cut, where o is the point of suspension, and c the point of oscillation; in other words, c is the point where a simple pendulum would produce the same oscillations as the compound pendulum. Compound pendulums are suspended either on a knife-edge, on the same principle as that of balances, or by means of a thin and flexible steel spring, which is bent slightly at every

oscillation.

In order to explain the oscillatory motion of the pendulum, we shall first notice the simple pendulum eм, fig. 16. When the material point м is below the point of suspension on the vertical passing through that point, the action of gravity is destroyed, or rather counteracted; but if the point be transferred to m, its weight P will be decomposed into two forces, the direction of the one being in the straight line em produced to B, and that of the other in the tangent m D to the arc m M n. The composant m B is counteracted by the resistance of the point e, but the composant m D urges the material point to descend from m to M. When it reaches this point, the pendulum does not stop; for, in consequence of its inertia, it proceeds in the

p....s

Fig. 16.

direction Mn. Now, if the same construction be made at any point of the are Mn, it will be found that the gravity which acted from m to M with an accelerating force will now act from M to n with a retarding force. It will take away, therefore, successively from the moveable the velocity acquired in its descent, so that, when it reaches the point n at a height equal to that of the point m, the velocity will become zero, as it was at the latter point. Whence it follows, that the same series of phenomena will be repeated, and the pendulum will continually oscillate. In practice, this result is prevented by the resistance of the air, and the rigidity of the cord, obstacles which can never be completely annihilated in compound pen

dulums.

Laws of the Oscillation of the Pendulum.-The passage of the pendulum from one extreme position or point m to the other is n called an oscillation or swing. The arc m n is called the amplitude of the oscillation; and the length of the simple pendulum is the distance of the point of suspension e from the material point M.

In treatises on Rational Mechanics, it is demonstrated that the oscillations of the simple pendulum are regulated by the four following laws.

1. In the same pendulum, the small oscillations are isochronous;

that is, that they are sensibly equal in the same time, so long as their amplitudes do not exceed a certain limit, namely 2° or 8o of the circle.

Galileo was the first who established the isochronism of the small oscillations of the pendulum. It is said that, when a young man, he first made this discovery by observing the motions of a lamp suspended in the dome of the cathedral at Pisa.

2. In pendulums of the same length, the duration of the oscillations are the same, whatever be the substances of which they are composed. Thus, simple pendulums of which the material point is composed of cork, lead, or gold, perform the same number of oscillations in the same time, if they are of equal length. 3. In pendulums of unequal length, the durations of their oscillations are proportional to the square roots of their lengths. Thus, if the lengths of pendulums he respectively 4, 9, 16, &c., times that of a given pendulum, the duration of their oscillations will be respectively 2, 3, 4, &c. times that of the oscillation of the given pendulum.

oscillations of a pendulum of the same length are in the inverse 4. At different places of the earth's surface, the durations of the ratio of the square roots of the intensities of gravity.

29

These laws are deduced from the formula t=π√ –, which is derived from the application of the calculus to the motion of the simple pendulum. In this formula, t denotes the duration of an oscillation; 7, the length of the pendulum; 2g, the intensity of gravity, that is, the velocity acquired at the end of the 1st second by a body falling in a vacuum. Also, is a constant quantity which denotes the ratio of the circumference of a circle to its diameter, which is equal to 3.141592.

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Between

Length of the Compound Pendulum.-The preceding laws and formulæ are applicable also to the compound pendulum; but in this case it is necessary to define what is meant by the length of the pendulum. Every compound pendulum is formed of a heavy rod terminating in a larger or smaller mass, according to its form and purpose; now, all the different points of such a pendulum tend, according to the third law of pendulum motion, to describe their oscillations in times differing from each other, and increasing in duration in proportion to the square roots of their distances from the point of suspension. But all these points being invariably connected together, their oscillations are necessarily performed in the same time. Hence, it is evident that the motion of the points nearer to the axis of suspension is retarded, and that of the points more remote from that axis is accelerated. these two extremes there are some points which are neither accelerated nor retarded, and which oscillate as if they were not connected with the rest of the mass. These points being all at the same distance from the axis of suspension, form together an axis of oscillation parallel to the former; now the distance of the axis of oscillation is called the length of the compound pendulum. Hence, the length of a compound pendulum is the same as the length of a simple pendulum which performs its oscillations in the same time. Thus in the preceding figure of the compound pendulum, the point o is the centre or place of the axis of suspen sion, and op the length of the compound mass; all the points of this mass between o and c are retarded, and all the points between p and c are accelerated; but all the points at o are neither accelerated nor retarded, and therefore the point c is the centre or place of the aris of oscillation.

the axis of suspension; that is, if we suspend the pendulum by The axis of oscillation possesses the property of reciprocity with its axis of oscillation, the duration of the oscillations will be the same as before; in other words, the axis of suspension will then length of the compound pendulum can be found experimentally. become the axis of oscillation. By means of this property, the This is done by inverting the pendulum and suspending it by such a manner that the number of oscillations performed in the means of a moveable axis, which is placed, after several trials, in

same time may be exactly the same as they were before its inversion. When this object has been attained, then the distance between the second axis of suspension and the first, is the true length required. If we now substitute the value thus obtained, instead of 1, in the formula relating to the simple pendulum, this formula becomes applicable to the compound pendulum, and the laws of oscillation are the same as those belonging to the simple pendulum.

as we have seen above, that gravity acts upon all bodies with the same intensity. They also enable us to determine the intensity of gravity at different points of the earth's surface, and consequently the true form of the earth itself. The isochronism of the oscillations renders it applicable as a regulator of timepieces. Lastly, M. Foucaud has recently employed it in the experimental demonstration of the diurnal rotation of the earth.

The length of the seconds pendulum, that is, the pendulum which beats 60 times in a minute, varies at every place, accord-dulum, we ascertain the value of 2g, from the equation tπ √ ing to the intensity of the force of gravity at that place: thus,

at the Equator, it is

at London, it is

39.0137 inches;
30.1393

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In order to measure the intensity of gravity by means of the pen2g By squaring both sides of this equation, we have f2=" 2g 27 at 10° from the Pole, it is 39-2106 whence, by reduetion, we have 2g= Thus, we see that, 12 Verification of the Laws of the Pendulum.-The laws of the in order to find the value of 2g at any place, we must measure the simple pendulum can only be verified by means of the compound length of the compound pendulum at that place, and then the pendulum; and this is best done by constructing the latter in duration of its oscillations; this may be found by ascertaining such a manner that it may fulfil, as much as possible, the condi- how many oscillations it makes in a given number of seconds, tions of the former; as, for instance, by suspending at the end of a and dividing the latter number by the number of oscillations. very fine thread, a small sphere of an extremely dense substance, such as lead or platinum. A pendulum of this construction different points on the earth's surface. Hence, by calculation, we By such experiments the value of 2g has been determined at Oscillates almost exactly like a simple pendulum, whose length is deduce from the value of 2g at each place, the distance of that place equal to the distance between the point of suspension and the from the centre of the earth, and consequently the form of the centre of the small sphere.

In order to verify the law of the isochronism of small oscillations, a pendulum of the preceding construction is made to oscillate, and the number of oscillations which it performs in equal times is noted when the amplitude is 3o, 2o, or 1°. By this means it is ascertained that the number of oscillations is in these cases exactly the same.

In order to prove the second law, several pendulums B, D, C, fig. 17, are constructed as suggested above; that is, having their Fig. 17.

B

earth itself.

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lengths in fine thread, equal and terminated in spheres of the same diameter, but of different substances, as lead, ivory, or brass. Neglecting the resistance of the air, it is found that all these pendulums make the same number of oscillations in the same time; whence it is inferred that gravity acts on all substances with the same intensity-a fact which has been formerly proved to the student.

The third law is verified by making pendulums oscillate, whose lengths are to one another respectively as the numbers 1, 4, 9, &c.; when it is found that the oscillations of these pendulums are to one another respectively as the numbers 1, 2, 3, &c. The fourth law, relating to the oscillations of pendulums, cannot be correctly proved by experiment.

Uses of the Pendulum. -The oscillations of the pendulum show,

Odoroso
Doloroso
Pomocotogno*
Tumultuo
Cuccurucu

Usufruttuo

dif-fee-tchee-lis-see-mee Very difficult

do-lo-ró-so

po-mo-ko-tón-nyo
too-móol-too-o
kook-koo-roo-kóo

oo-zoo-fróot-too-o

2. Words comprising the five vowels :—
Italian.
Affettuosi
Communicare

Pronounced.
ahf-fet-too-6-si
kom-moo-nee-káh-rai

Fragrant, odorous
Painful, dolorous
Quince

I excite a tumult

A word imitating
the cock-crowing
I have or enjoy the
temporary use

English.
Kind, affectionate
To communicate

• The sound of the gn will be explained in another lesson

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