Of, course, as before explained, when we speak of the power multiplied by the distance, we mean the product of the two numbers which represent the number of units of weight in the power, whatever they be, and the number of units of length in the distance.

In the case above let P p' be 5 inches, w w' will be 1 inch, and 3 (the number of pounds in the power), multiplied by 5 (the number of inches through which it has passed), is equal to 15 multiplied by 1. This equation, as it is called, is plainly true, each product being 15.

This principle, which is called the LAW OF VIRTUAL VELOCITIES, is the fundamental principle in mechanics, and holds good in all the mechanical powers. On account of its importance, it has been called the Golden Rule of Mechanics.

We will now trace its application to the wheel and axle, which is, in reality, only a modification of the lever, being an arrangement whereby an endless succession of levers may be brought into play, for any two radii of the wheel and axle in the same straight line may be considered as a simple lever.

If the power be slightly increased it descends, and, when the wheel has turned just once round, will have fallen through a space equal to the circumference of the wheel. In the same time, the weight will have been raised through a space equal to the circumference of the axle. But the circumference of circles always bear the same ratio to one another that their radii bear. If, then, the radius of the wheel be 12 inches, and that of the axle 3 inches, the power will pass through four times as great a space as the weight, but will only be one-fourth of it.

COMPOUND PULLEYS.

We are now in a position more clearly to understand the remaining mechanical powers. We have explained the principle of the simple pulley, and seen how to find out the advantage gained by it, both when the cords are parallel and when they are inclined at an angle. But there are various combinations of fixed and movable pulleys which are called compound pulleys, and are very frequently used in ships and in raising heavy weights, or exerting powerful strains. We must examine the principle of these, and see how to ascertain the advantages gained by using them.

They are usually classed in three systems.
In the first system,

2

B

4

E

which is represented in Fig. 72, each pulley hangs by a separate cord, and all are movable, at least all that have any effect, the runner, D, being introduced merely for the sake of reversing the direction in which the power acts.

Now let us try and see what is the ratio the power bears to the weight when the system is in equilibrium. We will suppose, for the sake of simplicity, that the power applied to P is 1 pound. The strain or tension of the cord PG is the same throughout its entire length (we are not taking into consideration now friction and the imperfect flexibility of the ropes). The pulley c is therefore kept at rest by the tensions of the three cords, GC, D C, and BC, and as these are parallel forces, the strain on B C is equal to the sum of the other two, each of which is 1 pound. The tension of each part of the cord c F is therefore 2 pounds. In the same way we see that the tension of the next cord, B E, is double that of CF, that is, 4 pounds. But the weight, Fig. 72. w, is supported by the two cords, B A, A E, or rather by these two parts of the same cord, and as each has a strain of 4 pounds, the total weight supported is 8 pounds. In this case then, there is a gain of 8, a power of 1 pound balancing a weight of 8 pounds. Similarly, if another pulley were added, a power of 1 pound would balance a weight of 16 pounds, each additional pulley doubling the weight supported; and thus we have the following rule for determining the gain in the first system of pulleys :

W

Multiply 2 by itself as many times as there are movable pulleys; the result will show the mechanical gain.

Thus, if there are five pulleys, the gain is 2×2×2×2×2, that is, 32. You must be careful, in calculating this, not to count the fixed pulley, as that has no effect.

In the second system of pulleys, instead of each having a separate cord, the same one passes round all, and they are arranged in two blocks, one of which is fixed and the other (usually the lower) is movable. Fig. 73 represents this system. One end of the cord is here fastened to the hook A on the fixed block, and it then passes in succession round the pulleys B, C, D, E, F, and G. In this case the weight is supported by six folds of the same cord, and each bears an equal part, the cord being equally strained throughout. Each part, therefore, sustains a portion of the weight equal to P, and w is therefore six times as great as P.

If we take away one pulley or sheave, as it is called, from the lower block, leaving two only, the weight will be divided between four folds of the cord, and thus only four times the weight of P will be supported. Similarly, were we to add another sheave to each block, we should have a mechanical advantage of 8. We see then that, in this system, the advantage is always twice as great as the number of pulleys in the movable blocks.

We have in this calculation supposed the cords to be parallel. They are not, however,

W

Fig. 74.

F

W

Fig. 73.

strictly so, still the difference
is so slight we need not notice
it. A trifling loss of power, however results from it.
Now there is one disadvantage about this sys-
tem when made as shown in our illustration, and
that is, that the weight must, on account of the
length of the blocks in which the pulleys are set,
be a long way below the point to which the
upper block is fixed. If we are using it, for
instance, to strain a telegraph wire or to tighten
a rope in the rigging of a ship, purposes for
which blocks are constantly employed, we should
fasten our rope to the hook from which w hangs,
and fasten the fixed block to the "dead eye" on
the side of the ship; but then we should not be
able to bring the rope within some considerable
distance of the eye. Another form of this same
system has therefore been contrived which ob-
viates this difficulty. This is shown in Fig. 74.

Here the required number of sheaves, three in the present case, are fixed side by side in each block, and the cord is fixed to a hook or staple in the upper one, and then passes in succession over each sheave. The weight is, of course, as before, divided between the six ropes. In all these cases it must be remembered that, as the lower block is suspended from the cords, it forms a part of the weight lifted, and the weight w is therefore less by this amount than our calculation makes it appear.

There is a third plan for arranging this system of pulleys which has the advantage of greatly reducing the amount of friction, there being only one sheave to turn on the axis instead of several. This will be understood from Fig. 75. Each sheave is here a compound one, as if several simple sheaves increasing in size were laid on each other. A little attention will show that for every inch the weight is raised 1 inch of cord will pass over the smallest pulley, and as the cord from this to No. 2 must also be shortened an inch, 2 inches will pass over No. 2. In like Fig. 75. manner 3 inches must pass over No. 3. Their sizes must therefore be in the proportion of the numbers 1, 2, 3, etc., or else the cord will grate on the pulleys. This is another illustration of our fundamental law; and we see further that, if the weight is to be raised 1 inch, each of the cords supporting

it must be shortened by that amount, so that in this case P must fall 6 inches to raise w 1 inch; but a power of 1 pound will balance a weight of 6 pounds; therefore, here also, the power multiplied by the distance through which it moves is equal to the weight multiplied by its distance.

These two systems are those in most general use. We must, however, just look at the third system, which is represented in Fig. 76. B is a pulley fastened to the cord B D, which passes over the runner A, and is made fast to the weight at the other end; c is likewise fixed to a cord which passes over B to the weight; a third cord passes over C, and P acts at the extremity of this. Each cord is thus fastened to the weight. Now in this case it is rather more difficult to find the advantage gained, as the weight is not shared equally by the three cords. The first cord is stretched by P, which, as before, we will call 1 pound; the part C F, therefore, supports 1 pound of the weight. The next cord is stretched by the tension of the two parts of the first, and its strain is therefore 2 pounds, which is the portion of the weight it sustains. Similarly the tension of the cord which passes over the pulley A is 4 pounds, and therefore the entire weight supported by P is the sum of 4, 2, and 1, that is, 7 pounds. Were another pulley added, P would in like manner support 8 + 4 +2+1, or 15 pounds.

P

24

FE

Fig. 76.

D

3. Three men are weighing an anchor with a capstan. Two have spokes 4 feet long, the third has one 5 feet 6 inches long. The radius of the axle is 6 inches. How much strain do they exert on the cable when each presses with a force of 100 pounds?

4. A man with a winch 18 inches long turns a pinion with 6 teeth, this works in a wheel of 60 teeth, carrying another pinion with 8, which drives a wheel with 54; round the axle of this the cord passes. the radius of the axle be 4 inches, what force must be exerted to raise a weight of 2 tons?

If

(The radii must here be considered as proportional to the number of teeth.)

5. In the simple pulley (Fig. 69), if the angle a wb is 90 degrees, and the power is 49 pounds, what weight can be supported?

6. In the second system, when there are four movable pulleys, how many pounds must I pull with to overcome a resistance of 7 hundred

weight, and how far must the power move to raise the weight 6 inches? 7. In the third system, with 5 pulleys, what weight will 5 pounds support; and how strong must the middle rope be?

8. If a cask weighing 150 pounds be suspended from a pole 8 feet long, and carried between two men on their shoulders, if the point of suspension be 3 feet from the front man, how much of the weight will each bear?

GENITIVE.

LESSONS IN CERMAN.—XXV. SECTION XLVII.-ADJECTIVES REQUIRING THE ADJECTIVES in German that require the genitive, generally answer to our adjectives followed by the preposition "of" ($ 124), ich bin der Sache gewiß, I am sure of the matter. Er ist seiner nf, he is mindful of his duty. ie fine teiner Freund, they are unworthy of thy friendship. They are rendered by adjectives followed by the preposi"to," as:-Er ist res Weges kuntig, he is acknowledge of) the road. Es ist nicht der Mühe

orth the trouble.

Some of these adjectives are also used with the accusative (§ 124. 2. Obs.), as :-Ich bin das Gehen nicht gewohnt, I am not accustomed to walking. Gr ist es müde, he is tired of it. VOCABULARY.

Fremin,devout, pious. Schuld, f. fault, guilt.
Geinig, covetous.
chuldig, guilty.
Geschwäß', n. gossip., Cumme, f. sum.
Gekan vig, confess. That, f. deed.

ing.
Handwerk, n. handi-

craft, trade. Haut f. skin, hide. Helfen, to help. Kundig (sein), (to be) acquainted (with). Lantgut, n. farm, estate. Lant leben, n. country life.

Nachste, m. neighbour. Niemals, never, at no time.

Freundschaft, f. friend- Richter, m. judge.
ship.
Schießen, to shoot.
RÉSUMÉ OF EXAMPLES.

Die ter Keßerei' an'geklagten Per.
so'nen mußten in der Berzeit ten
Feu'ertet erleiden.

Der Arme, ter einer Sache bedürftig ist, ist gewöhnlich auch einer an dern benöthigt.

Wenn das Pfert sich seiner Stärke bewußt wäre, könnte es Niemant bantigen.

Nur derjenige, welcher tüchtig in seinem Beruf ist, kann seines Lebens froh werten.

3ch bin meiner Sache gewiß und des Weges kundig.

n das Fahren gewöhnt', werte ich

balt des Gehens müre.
Schulrig over nicht schuldig eines
Verbrech ́ens, jeder wurte ver-
ur theilt.

Der Jäger ist seines Zieles sicher.
3ch bin jezt meines Lebens über.
trüffig.

Der Dieb ist des Diebstahls, der
Mörter des Mortes, und in einem
respo'tischen Lande, eine frei'beits.
liebente Seele des Hoch'verraths

schuldig.

Mancher Mensch verbringt sein Le ben un'einget enk seiner e'wigen Wenn er nur seiner Schuld gestän'. Bestimmung. rig wäre, ich wollte ihm gern ver zei'hen.

bron, m. throne.

Tüchtig, well quali

fied.

quainted with. Unwerth, unworthy. Verdächtig suspected. Vertrin fen, to spend

(for drink). Verü'ben, to commit. Werth, worthy. Zukunft, f. future.

The persons accused of heresy were compelled in (the) former times to suffer death by fire.

The poor (man) who is destitute

of one thing is commonly also (needy) in want of another. If the horse were conscious of

his strength, none could con trol him. Only he (the one) who is well

qualified for his calling can I am sure of my cause, and ac(be glad cf) enjoy his life. quainted with the road. Accustomed to ride, I soon become tired of walking. Guilty or not guilty of a crime, every one was condemned. The hunter is sure of his aim. I am now weary of my life.

A thief is guilty of (the) theft, a murderer of (the) murder, and, in a despotic country, a liberty-loving soul of high treason.

Many a man wastes his life unmindful of his eternal desIf he would only confess (were tiny. confessing, etc.) his guilt, I would gladly pardon him.

EXERCISE 90.

1. Ich bin einer solchen Arbeit ungewohnt und würte sie nicht thun, wenn ich nicht res Meltes betürftig wäre. 2. Ich bin einer großen Summe Geltes benöthigt; helfen Sie mir, ich bin Ihrer Hülfe gewiß nicht unwerth 3. Wenn er eingerenk meiner Wohlthaten wäre, würte er nicht so hanteln. 4. Dieser Mensch ist so schlecht, daß ich ihn jeter That fähig halte. 5. Glauben Sie, daß ein Geiziger seines Lebens froh sein kann? 6. 3ch will ja gern meiner That geständig sein, laß mich nur laufen! 7. Der Jäger was seiner Beute so gewiß, daß er die Haut des Bären vertrank, bevor er ihn ge schossen hatte. 8. Laß mich jezt gehen, ich bin deines Geschwäßes berzlich müre. 9. Gut, wenn Sie meiner überbrüssig sind, werte ich gehen. 10. Niemals werte ich mich einer That schuldig machen, welche mich Ihrer Freundschaft unwürtig machen würte. 11. Ich befize ein Lanrgut, aber ungewohnt der Arbeit, und unkuntig res Ackerbaucs, bin ich desselben über, trüg. 12. Einer ift des Andern werth, aber auch oft Einer des Andern unwerth. 13. Ein König, ter vas Volk nicht liebt, ist des Thrones un

die gefährlichsten, welche ihm nicht bemerkbar sind.

Sei deinen Freunden ergeben, und deinen Feinden gefällig.

EXERCISE 91.

1. America has a golden soil for him who is acquainted with a trade. 2. The covetous man is insensible to the misery of others. 3. The Italians who rose in some provinces against Austrian oppression were unmindful of their weakness. 4. I became tired of hearing such a long speech. 5. Everybody is sure of death, but no one is always mindful of it. 6. If thou wert so accustomed to country life as I am, thou wouldst not stay in town. 7. Never will I make myself guilty of a deed which will make me unworthy of your friendship. 8. The man who has led a conscientious life, and followed the dictates of his conscience, has no fear of dying; but the wicked man, unmindful of his actions and conscious of his crimes, is afraid of death and the future. 9. Many a one who stands accused of a crime before his judges is conscious of less offence than those who judge him.

SECTION XLVIII.-ADJECTIVES REQUIRING THE DATIVE. Adjectives in German that require the dative usually answer to our adjectives followed by the preposition "to," as:-Es ist mir angenehm, it is agreeable to me. Er ist seinem Vater ähnlich, he is similar to (resembles) his father. Er ist seinem Herrn ergeben, he is devoted to his master. Das Geschäft ist ihm lastig, the business is disagreeable to him. (§ 131.)

able. Duraus', throughoat, entirely. Gregben, devoted. Erigen, to heat, in

flame. Erinnerlich, coming

to mind. Griteter, former. Enig, eternally. Scine, hostile.

Neigung, f. inclination.

O'brigkeit, f.authority. Pfat, m. path. Rauchen, to smoke. Schärlich, injurious. Schmeichelhaft,

flat

tering. Schwach, weak, infirm. Spiel, n. play. RÉSUMÉ OF EXAMPLES.

Beter rechtliche Mann ist rem Gu ten geneigt und tem Bösen ab', geneigt.

Er ist nichts mehr an'gelegen, als meinen Freunten behülflich zu fein wo ich kann.

Der Hund ist seinem Herrn gehorsam unt tankbar; warum will ter Mensch ihm nicht gleich sein?

ln'terschied, m. distinction.

Un'terthä'nig, submis

Was dem Einen nüglich ist, ist oft dem Andern schädlich.

Ich bin Ihnen für Ihre Güte sehr verbunden.

Manche Menschen sind den stillen Wassern gleich, an'genehm dem Auge, aber untreu dem'jenigen, der sich ihnen anʼvertraut. 3ch werte mir se Sache auch ohne teine Erma'aungen an'gelegen sein lassen.

To the sailor, those rocks (cliffs) are the most dangerous which are not perceptible to him. Be devoted to thy friends, and obliging to thy enemies. What is useful to one is often injurious to another.

I am very much obliged to you for your kindness. Many men are like the quiet waters, agreeable to the eye, but unfaithful to him who entrusts himself to them.

I shall take this matter to heart (let it be important to me), even without your exhortations.

EXERCISE 92.

1. Angenehm wäre es mir, wenn ich einmal einen Mann fände, der mir 2. Wenn Sie mir auch verwandt sind, so scheint mir doch anhängig wäre. Ihr Betragen durchaus nicht anständig, und ich rächte, es wäre Ihnen felbft bemerkbar, daß Leute, tenen Ihr Betragen bekannt ist, Ihnen nicht gewogen sind. 3. Meinem Vater ist die lezte Theuerung noch sehr erinnerlich. 4. Gefährlich ist dem Wanderer jener Pfad. 5. Sei gefällig deinem Nächsten, gehorsam teiner Obrigkeit, dann wird dir diese geneigt Daß der Fürst tem Volte nicht gleichgesinnt, und das Volk dem Fürsten und jener gewogen sein. 6. Ist Ihnen das versprochene Geld gewiß? 7. gleichgültig ist, macht dem Erstern das Regieren schwer, und ist dem Leßtern hinterlich an seinem Glücke. 8. Die Sterne sind mir günstig, leicht wird mir mein Unternehmen sein. 9. Wenn die Wanzen dem Menschen auch nicht schädlich sind, so sind sie ihm voch lästig. 10. Mir ist es sehr lieb, daß ich tir in dieser Sache nüglich sein kann. 11. Meinem Freunde war Ihr Lob sehr schmeichelhaft. 12. Gut zu werden, ist dem Lasterhaften schwer, denn er bleibt gewöhnlich seinen Neigungen treu. 13. Mancher Schwache ist dem Starken an Geist überlegen. 14. Welch ein Unterschied ist zwischen: ein Mensch ist dem andern ungleich, und ein Mensch ist dem antern unähnlich? 15. Wie steht das Spiel? 16. Mir sehr ungünstig. 17. Se unlieb es mir ist, muß ich dir erklären, daß mir dein Geschwäß unLeiblich ist. 18. Wer möchte wohl den Grundsaß glauben: „wer seinem König nicht unterthänig, ist untreu seinem Vaterlande?" 19. Es ist mir ewig unvergeßlich, wie sehr ich dir verbunden bin. 20. Nicht jeder, der mir verwandt ist, ist mir auch wohlgefällig. 21. Was mir angehörig ist, lasse ich mir auch angelegen sein.

EXERCISE 93.

1. Who can be hostile to a child? 2. Is it agreeable to you to take a walk ? 3. This is injurious to your business. 4. Everybody has been favourable to him. 5. A benevolent action is well-pleasing to God. 6. You are very similar to your father in your habits, for he was disinclined to smoking and averse to drinking. 7. What belongs to me, that I shall also take to heart. 8. To every thinking man it is observable, that it is not easy for a prince to make the people devoted to him. 9. With pleasure will I be serviceable to you, in obtaining a situation. 10. Be obliging to thy parents, obedient to their authority; then will they be inclined to thee, and favourable to thy prosperity. 11. To drink cold water is injurious to a heated body. 12. The horse is a docile animal, and obedient to his master. 13. If it be agreeable to you, come to my house tomorrow, at noon. 14. The dog is submissive and faithful to his owner. 15. He was inclined to make himself disagreeable to

portant to me than to be serviceable to my friends where I can.

The dog is obedient to his master, and thankful; why will man not resemble him?

good girl's new comb. 11. Has the old watchmaker's young friend the old servant's beautiful horse? 12. No, he has the rich Englishman's horse. 13. Have you the good tailor's dress-coat ? 14. No, I have this new dress-coat from the good tailor. 15. Have you this poor weaver's cloth ? 16. No, I have cloth from the weaver. all old wine strong ? 18. No, and not all new wine is weak. new dress-coat is of black cloth,

17. Is 19. The

LESSONS IN ARCHITECTURE.-IV.

PROGRESS OF ARCHITECTURE IN EGYPT AND INDIA-THE

COLUMN.

AFTER the rude style of building practised in early times had spread itself in various forms over the ancient world, true art at last made its appearance. The great nations of antiquity, as they advanced in civilisation, created a national architecture, each with a feeling and expression peculiarly its own. As soon as the stones used in buildings began to assume a cubical, prismatic, or cylindrical form, and the square and compasses gave a new direction to stone-cutting, architects gave wing to their imagination, because they now had the means of realising its creations. Symmetry was

A

question

studied in the ground-plan of their edifices; their
architraves were raised upon pillars and columns;
and experience ere long taught them the strength
of every stone, and the proper height of every
part of a building. Hence arose that harmony and
proportion which
elevated architec-
ture into an art.
We shall not at-
tempt to decide the

whether pillars and columns were first formed in excavations, or in separate constructions; but it is evident that they were the first elements of a regular architecture-that is to say, of the orders which constituted the first basis of architectural harmony. To the pyramidal constructions of Egypt and of Asia speedily succeeded the erection of palaces and other edifices, in which square and cylindrical pillars formed a most essential part: the great weight of the materials employed requiring that they should be supported at short distances for the formation of internal and external galleries. These single pillars could only be connected at the top by architraves of such dimensions as combined the ratio of their breadth with the proportions of the supporting power of the columns. Upon these architraves were placed platforms or ceilings of flat stones, which, by their thickness, formed a new di

The most ancient monuments of Egypt ornamented with columns are situated in the Heptanomis, an old division of the country which corresponded pretty nearly with the district called Middle Egypt by modern geographers, and which was situated between the Delta and the Thebaid, extending from 24° N. lat. to 30° N. lat. These monuments exhibit speci

mens of the greatest simplicity, and strongly analogous to those of the Doric order. The monuments of India excavated in the rock present the same principles of these primitive orders. In these two countries, which are the cradles of architecture, artists at first decorated their columns and their capitals with ornaments of which the ideas were taken from the local vegetation, to which were sometimes added others borrowed from animal nature. Thus in Egypt, after having set up the simple cylindrical shaft for their column, they sculptured upon it branches of the lotus, meeting each other and fastened together by fillets. The capital which crowned the column was at first composed

mension above the former; and upon these platforms were formed terraces or flat roofs, which were surrounded by another row of stones forming a border, and having an outward projection which preserved the façade from the effects of the rain. These were the origin of cornices and entablatures. The column, in preference to the square pillar, became the type of architectural proportion. Simple at first, it presented nothing but a cylindrical shaft, without ornament, and only expressing the purpose for which it was originally intended. The oldest specimens in Egypt are of this description; Asia presents similar specimens; and Greece, with the whole of the West, follows the same track: thus proving that everywhere there is an invariable similarity in the origin of the arts. The simplicity, elegance, and utility of the column engaged the attention of architects, and concentrated all the efforts of their imagination. Thus it became their architectural type or model, and formed the nucleus of the different characteristic styles of building that were adopted by the great nations of antiquity.

VOL. II.

of the bud of the same flower. This first idea was afterwards developed in the application of

vegetation of every kind to the ornamentation of the columns of the temples and of the great public edifices. Among the six examples of Egyptian capitals given in this page there is one composed of the leaves of the palm-tree.

Egypt, thus adorned with orders of architecture, had its national style. The numerous works upon the history and antiquities of Egypt published during the last half-century have made us acquainted with its archæological treasures, such as the temples and palaces of Thebes, the Isle of Philæ, Karnac, Abousambul, Edfou, Memphis, and others; and large public buildings, decorated with numerous columns, immense pil

lars, obelisks, and sphinxes, which give to this style a peculiar character of antiquity and grandeur, of which mere verbal description would fail

to convey any idea to the reader.

In India, as in Egypt, iso

lated columns and pillars appear to have had their origin in subterranean excavations for architectural purposes; of these numerous examples are seen at Ellora, in the palace or temple of Indra. These pillars are much shorter than those of Egypt, their bases and capitals occupying a considerable portion of the height of the column, and the entablature, or rather the corona, is less accurately traced. In cases where the Indians cut out the rock for the purposes of decoration, and sculptured it over with various ornaments, the column assumes a lighter appearance, and the principle of an order of architecture can be traced.

The excavated temples of India are numerous and extensive; the principal ones are those of Elephanta, Salsette, and Vellore, or Ellora. Elephanta is situated near Bombay, on a small island of the same name, which received this appellation from the figure of an elephant being cut out upon the rocks on the southern shore. The grand temple is 120 feet square, and is supported by four rows of pillars; along the side of the cavern are fifty colossal statues from twelve to fifteen feet 31