follows. The distances e d, a c, cb, are each equal to one-fourth of an inch, and da to three-sixteenths of an inch.

By following the above instructions, the learner will be enabled to rule his paper in sets of five horizontal and parallel lines, five lines being required in each set, in this case, to determine the height of the letters and their relative proportions one to another, To rule the diagonal lines (No. 5), set off e xalong the topmost line ce, equal to thirteen-sixteenths of an inch, and draw a straight line through the points br. This will serve as your guide line for regulating the remainder of the sloping lines, and all that remains to complete them is to set off spaces along the lines ee, bb, equal to x y or b z, and rule straight lines passing through every succeeding pair of points, commencing from the first pair a b, through which the guide line for regulating the

inclined lines was drawn.

LESSONS IN GERMAN.-III. SECTION V.-THE NOUN. OLD DECLENSION. THERE are in German four cases, namely: the Nominativ, | answering to the English nominative; the Genitiv, answering to the English possessive; the Dativ, which has no corresponding case in English; and the Accusativ, which answers to the English objective.

Of the four cases, the dative, without a preposition, generally

corresponds to our objective governed by to or for, as :-
Ich gebe tem Manne das Glas. I give (to) the man the glass.
Er macht dem Manne einen Hut. He makes (for) the man a hat.
Often, however, the dative in German is construed with a

preposition, where, as above, the objective is of course employed

in English, as :

Das Kind ist in dem Hause,

Der Hund ist unter dem Baume. Der Jäger geht nach dem Walde. Der Mann ist auf dem Schiffe. Der Koch ist an dem Tische.

The child is in the house.
The dog is under the tree.
The hunter goes to the forest.
The man is on the ship.
The cook is at the table.

Nouns ending in 8, F, 3, or two consonants, generally add es in the genitive, thus, like our words which end with the sound of s, 2, z, soft c or s, forming an additional syllable.

Nom. Das Ros, the horse. Gen. Des Rosses, the horse's.

RULES FOR FORMING THE CASES OF NOUNS ACCORDING TO
THE OLD DECLENSION.

RULE I. The genitive adds 8 or cs to the nominative.
RULE II. The dative drops the 8 of the genitive (§ 13. Note).
RULE III. The accusative is like the nominative.

DECLENSION OF NOUNS ADDING IN THE GENITIVE.

An, at.
Auf, on.
Ball, m. ball.
Bruter, m. brother.
Durstig, thirsty.
Freunt, m. friend.
Hunt, m. dog.
Hut, m. hat.

The father of the child gives (to) the brother the bird. The ruler of the state sends (to)

the warrior the sword.

VOCABULARY.

In, in.
Korb, m. basket.
Loben, to praise.
Pfert, n. horse.
Sad, m. bag.
Schläfrig, sleepy.
Schüler, m. scholar.
Sein, to be.

RÉSUMÉ OF EXAMPLES.

Der Wolf lebt in dem Walte.
Das Kind liebt ren Bruder.
Der Lehrer lobt res Schülers Fleiß.

Der Schnee liegt auf tem Verge. Das Märchen hat des Vaters Hut. Der Sohn des Bäckers hat Brod in tem Korbe; er giebt es dem Bettler.

Sohn, m. son.
Stall, m. stable.
Stuhl, m. chair.
Tischler, m. joiner.
Unter, under.
Wo? where ?
Zimmer, n. room.
Zimmermann, m. car-
penter.

The wolf lives in the forest. The child loves the brother. The teacher praises the scholar's industry.

The snow lies on the mountain. The girl has the father's hat. The son of the baker has bread in the basket; he gives it to the beggar.

EXERCISE 6.

1. Sind Sie der Freund des Bäckers? 2. Nein, ich bin der Freund des Tischlers. 3. Was hat der Freund des Fleischers? 4. Er hat den Hund und das Pferd tes Bauers. 5. Wo ist das Mehl? 6. Es ist in dem Sade des Müllers. 7. Wo ist tas Korn? 8. Es ist in dem Korbe des Vauers. 9. Wer licht den Lehrer? 10. Der Schüler liebt den Lehrer. 11. Sie Sie schläfrig? 12. Nein, ich bin durstig. 13. Wo ist der Vall des Bruders? 14. Das Kind hat den Ball des Bruders in dem Hute des Vaters. 15. Wo ist das Pferd des Lehrers? 16. Es ist in dem Stalle. 17. Lobt der Tischler den Zimmermann? 18. Nein, ter Sohn des Zimmermanns lobt den Sohn des Lehrers. 19. Wo ist der Stuhl des Tischlers? 20. Gr (Sect. XVIII. 3) ist in dem Zimmer des Lehrers. 21. Liebt der Zimmermann den Lehrer? 22. Ja, er liebt und lobt den Lehrer. 23. Der Mann ist an dem Tische, das Buch ist auf dem Tische, und der Hund ist unter dem Tische.

SECTION VI.-DEMONSTRATIVE PRONOUNS. Dieser" is declined, in the masculine, precisely like the definite article; while in the neuter, as will be seen in the following declension, all its endings, except the dative, are alike

Ich bin, I am;

Sie fint, you are;

Grist, he is;

ich lobe, I praise; Sie leben, you praise;

er lobt, he praises.

By the references in Roman numerals, thus (Sect. XVIII. 3), as above, the learner is directed to Sections in Part I. of theso Lessons. References thus [§ 62 (2)] refer to the Sections in Part II.

reverse the operation by multiplying by that number. Thus, we ree that "ton," "hundred-weight," and "pound," are only so many different expressions for the same unit-namely, the pound either singly or collectively, and that, therefore, for practical purposes, we may say that a pound weight is the "unit of force."

But we cannot leave this subject without determining the relation between this unit and the very small one of which I first made mention. I have asked you to take it on credit that the latter is nearly eight grains. The more correct value involves decimals, and is 7.85 grains nearly, that is, seven grains and eighty-five parts out of a hundred of one grain. Hence, since there are 7,000 grains in an avoirdupois pound, if we divide this number by 7.85, we shall have the number of these small units (which henceforth we shall call the dynamical mit), to which one pound weight is equal. The division gives 892 nearly for the quotient; and thus we learn how we may pass from dynamical units to pounds, or from pounds to these nnits. The result may be summed up in the following table :7.85 Grains make nearly one Dynamical Unit. 892 Dynamical Units make nearly one Pound. 112 Pounds make one Hundred-weight. 20 Hundreds make one Ton.

:

Forces applied to a Point. When a single force is applied to any point of a body, if the latter be free, motion will ensue, and the question belongs to Dynamics. If it be not free, but fastened in any way to fixed objects, the force will be communicated through its substance to the points of support or connection, which will resist, and by resisting cause the body to sustain strain. For example, suppose a beam of wood is fixed at one point, round which, as on a pivot, it can turn in any direction, and that a force is applied to it at some other point. It is clear that this force will pull the beam round towards itself so far as it can go, that is, until the line of direction of the force passes through the fixed point. Then this point will resist, and equilibrium will be produced. The case thus becomes one of two forces-namely, that applied and the resistance produced; and we see thus that a single force can never in Statics be the subject of study, without involving the consideration of other forces which it calls into existence. A statical problem must be concerned about at least two forces.

If two forces be applied to a point in the same direction, we assume in Mechanics, as a self-evident truth, the result of experience, that their joint effect is the same as that which would be produced by a single force equal to their sum. If two men of unequal strength pull on a rope against another man stronger than either, who succeeds in balancing their united strength, we say, without hesitation, that his force is equal to the sum of those put forth by the two. When two forces thus act separately at a point, the single force to which their joint power is equal is called the "resultant" of these forces. We therefore say, if two forces act on a point in the same direction, their resultant is the sum of these forces. If three act on it, since two of them are equivalent to one equal to their sum, this one with the third must be equivalent to a single force equal to the sum of the three. And so on, as to more than three, we may lay it down as a general rule that—

The resultant of any number of forces acting on a point in the same direction, is a single force equal to the sum of the separate forces.

When two forces act in opposite directions on a point, for the same reason as in the former case, we assume that the resultant is the difference of the two. And this leads us to the most general case that can occur of such forces-namely, that in which any number of them are applied to a body along the same line, some in one direction and others in the opposite direction. To determine the resultant of all, it is evident that it is sufficient to take the separate resultants of the opposing sets, then take the difference of these resultants, and that this difference will be the required resultant of all, and its direction that of the greater of the two separate resultants. Hence the following rule:

If any number of forces be applied to a body along the same line, their resultant is the difference between the sums of those which act in the opposite direction, and its direction is the same as that of the greater sum.

forces applied to the rope along its length is the difference between tho united powers of the fifteen and of the eleven, whatever be the particular strength of each man, and its direction is that in which the fifteen pull.

But suppose now that two forces only are employed, and that they are equal and in opposite directions; what will be the result? They will balance, or be in equilibrium. Now it is sometimes said that the body to which two such forces are applied at one of its points is in the same condition as if no force had been applied to it. This is not true, strictly. It is in the same condition so far as equilibrium is concerned, but not otherwise. It is not in the same condition as to pressure or strain. The rope, which at one moment is lying stretched on the ground, is not in the same condition it was in a few minutes before, when two strong men were pulling at opposite ends of it with balanced strength. In the latter case it is strained along its whole length-every thread on the stretch, ready to snap. Its condition is very different on the two occasions-different in every circumstance, except that of there being no motion. So, also, if two equal and opposite pressures are applied to a round ball, it will be an equilibrium, but the condition of its substance will be changed. Its particles will be pressed towards one another inwards; and, if it be made of soft or elastic material, its form will be altered by the flattening effect of the opposing forces. And this is true, whatever be the magnitude of the ball. It may be as small as we please, even so small as an atom, or what is called a "material particle," and yet there will be this internal compression or straining. Thus we see that even the "material particle," acted on by two equal and opposite forces, cannot be said to be in the same condition before and after their application.

The case of equal and opposite forces presents some other points of interest, which may well occupy your attention in this lesson. Suppose, for example, two men pull against each other with equal strength at the opposite ends of a rope. What will be the strain on the rope? What will be its amount, considering that both are pulling? Most persons at first incline to say that it is strained by the united strength of both, or by double the strength of either man. Such is not the case; the strain is only equal to the strength of one of the men. What is the reason of this? A moment's reflection makes it evident. Suppose one man only to pull; the rope follows him, and there is no strain on it. But the instant the other scizes his end and pulls, strain begins, caused by his resistance. If he gives a strong pull, it is great; if a weak, it is slight. But, to put this in another way, suppose the first man leads, pulling with all his might, while the other, holding on with less strength, is dragged after. The rope is strained in this case also. By how much? By the loss of the two forces. The stronger pull becomes divided into two parts, one putting both the rope and the second man in motion, and the other balancing the latter's pull. It is this second portion which strains the rope, and must be equal to the strength of the hinder man, while the other, which causes motion, is the difference of the two pulls or forces. Suppose, lastly, that the two pulls become equal, their difference becomes nothing, motion ceases, and the men come to a standstill. But the strain remains, as before, equal to the hinder force, which, being equal to that of the leading man, we can say it is equal to either of the forces.

Let us next suppose that for one of the men an iron ring, fastened on a wall, is substituted, to which one end of the rope is attached. So long as the rope hangs loosely from the ring there is no strain on it. Let the other man now pull at the far end, the rope at once is strained, evidently not by the wall, but by the man's pull. The wall puts forth no more effort to strain it than it did before; but simply resists the force communicated to it through the rope. It is, in fact, a case of a force applied to the wall through the rope, every point of which may be considered a point of its application.

Again, take two equal weights attached to the ends of a cord which passes over a pulley. The strain on the cord which hangs down at either side is evidently equal to the weight on that side; and, since the weights are equal, the strains on both sides, and therefore all through the cord, are equal to that weight.

the bullocks overcomes the weight of the water and bucket, and an amount of motion results, due to the difference of the two forces. The rope, however, is strained only by the weaker force, evidently so in the part which descends from the pulley to the bucket, and therefore also in the remainder, since the strain must be uniform along its whole length.

In all these cases the forces were of the nature of a pull, causing a stretching strain. But the conclusions hold equally good of pushing forces. If two such, equal to each other, be applied to a ball at opposite sides in opposite directions, the compressing strain within the ball will be equal to only one of the forces. Or if the ball be pushed against a wall by only one of them, though the wall resists, the strain I will still be the same equal to the single force. The resistance counts for nothing. Also, when the two forces are unequal, and motion ensues, there is a compressing strain equal to the smaller force, while the motion produced is due to the difference of the forces. When a man ascends a ladder with a hod of mortar, there are two such compressing forces acting on his shoulder at the spot on which the hod rests-namely, his own muscular power pushing his shoulder up

wards, and the weight of the hod and mortar pushing it down. His ascent is effected by

the difference of these forces, the muscular being the greater; while the compressing strain is evidently the weight of the loaded hod. These examples will make clear to you the principle I have been explaining; and you will find no difficulty in multiplying them by thinking of others yourselves.

We now pass to the case of three forces, whose directions are all different, applied to a point, and producing equilibrium. Now it is

-evident, first of all, that mill the three must pull or push in the same plane or flat, such as, for instance, the flat surface

other, and therefore could not make equilibrium. In the case of the ring on the table, to which the three strings are attached, if the direction of the effect of the pulls on two of the strings were not opposite to that of the third pull, the three would make the ring move to the side of the table, towards which these two directions incline. And, furthermore, even if the directions were opposite, the ring would move, if the effect of the two, or their resultant, were not equal to the third force. These two principles may be definitely stated as follows:1. When three forces applied to a point are in equilibrium, they are in the same plane.

2. The resultant of any two of three forces in equilibrium at a point is equal and opposite to the third force. From these principles it is evident that in order to ascer

of a table; for if two of them pulled along that surface, while the third pulled in a slanting direction upwards, this latter force should lift the body off the table. Try the experiment with three strings attached to a ring which lies flat on a table, two of which are pulled horizontally along the table, and the third in any direction upwards. The ring will be lifted, and soon the three strings will come into one plane. I am not here taking into account the weight of the ring and strings, which are a fourth force applied to the body. For the sake of simplification, to enable you to understand the principle, I suppose these to be so small in comparison to the others as to count for nothing. Secondly, when three forces to a point are in equiliis equal and opposite if it were not, the which the three are l opposite to each

brium, the resultant of

to the third force. T

resultant of the t

-equivalent, wo

7

to hang down in the middle, we shall have a case of three equal forces applied to a point. There are the two outside weights acting over the pulley, and drawing the knot obliquely to either side, and the middle weight pulling it downwards. What position will the strings settle themselves into? Evidently so that the angles all round between the strings may be equal; for no reason in the world can be given why they should be unequal. Whatever reason could be assigned for supposing one of these angles greater than the other, since the forces are equal all round and all the other circumstances the same, that same reason should make that other angle greater than the first. The angles, therefore, must be equal. Let any one of you make the experiment, and measure the angles, and he will find the result to be as I have stated. But you will find this same conclusion arrived at in the next lesson in another and more satisfactory manner, by the Parallelogram of Forces.