longer arm. Thus arranged, you will exactly balance each other; and as each of you, on your descent, touches the ground with your feet, the reaction affords you a spring, which destroys the equilibrium, and enables you to oscillate in arcs about the centre of motion."

"Do we then describe the arcs of a circle as we ascend and descend?"

"Undoubtedly you must. Look at this diagram," said Mr Seymour, "and you will see at once that the plank can only move round its centre of motion; for how could you rise, or your brother fall, perpendicularly in a straight line? You must, in rising, and he, in descending, describe arcs of your respective circles. It is equally evident that his velocity must be very superior to yours; for, if you could swing quite round, you would each complete your respective circles in the same time."

"It would really appear so," said Tom; "and I have myself observed that the lighter person has the better ride, as he moves both farther and quicker, and I now understand the reason of it; it is because, being farther from the centre of motion, he describes a larger arc."

"The greater velocity with which your little brother moves, renders his momentum equal to yours. You have the most gravity, he the greatest velocity; so that, upon the whole, your momenta are equal: for you, no doubt, remember that momentum is weight multiplied into velocity.* You have here, then, a striking instance of mechanical advantage gained by opposing motion to matter, or velocity to weight; for I think you will readily admit, that, without the aid of the plank, your little brother could never have raised you from the ground." "That is clear enough," said Tom.

"The plank, then, thus arranged," continued his father, "constitutes what has been termed a mechanical power, to which the name of lever has been given; it is not, however, my intention at present to enter into the history of these powers, of which there are six distinct kinds; the one presented to you, in the instance of the see-saw, is perhaps the most simple, and not the least important of them."

"It is very curious," observed the vicar, "to reflect upon what a simple, and apparently trifling fact, the powers of civilized man may be said to depend. The single truth you have just announced, of making velocity a compensation for weight, has supplied his weak arm with the means of controlling the very elements."

"It is very true," said Mr Seymour; "and we might go so far as to say that, had it been the will of the Almighty Creator of the universe to have withheld from matter that property which we have been discussing, man must have remained the most helpless and forlorn of His creatures. I now propose," added Mr Seymour, "to accompany * See page 47.

the children to their swing; the present is a suitable opportunity for giving them some idea of the doctrine of oscillation, or the theory of the pendulum."

The children had commenced the sport, and Mr Seymour informed Tom and Louisa, who were attentively watching the motions of the swing, that its vibrations, like those of the pendulum of a clock, were produced by its effort to fall, from the force of gravity, and its power of ascending through an arc similar and opposite to that through which it had descended, from the momentum acquired during its descent. "Like the bandalore, I suppose," said Louisa.

"Exactly, my dear, that is a very good comparison; for as the bandalore, having descended along the string by its gravity, acquires such a momentum as to enable it to ascend the same string, and thus, as it were, to wind itself up; so does the pendulum or swing, during its descent, acquire a force that carries it up in an opposite arc to an equal height as that from which it had fallen. But tell me, Tom, whether you have not discovered that the motion of your new swing differs from that which you experienced in your former one?"

"The ropes of our present swing are so much longer than those which we formerly used, that the motion is much pleasanter."

"Is that all?" said Mr Seymour. "Have you not observed that you also swing much slower ?"

"I have certainly noticed that," said Tom.

"It is a law which I am desirous of impressing upon your memory, that the shorter the pendulum, or swing, the quicker are its motions, and vice versa: indeed, there is an established relation between the velocity and the length, which I shall hereafter endeavour to explain to you. Galileo, the celebrated philosopher, and mathematician to the Duke of Florence, accordingly proposed a method of ascertaining the height of the arched ceiling of a church by the vibrations of a lamp suspended from it. The solution of the problem was founded on the law to which I have just alluded, but which involves mathematical considerations, with which it is not my present intention to perplex you. Now it is known that, in the latitude of London, a pendulum, if 39 inches and two-tenths in length, will vibrate seconds, or make 60 swings in a minute; by observing, therefore, how much the pendulous body deviates from this standard, we may, by the application of the appropriate rule, find its length; if the distance from the bottom of the lamp to the pavement be then measured, which may be done by means of a stick, and added to the former result, the sum will give the height of the arch above the pavement; but I will shew you the experiment the next time we go into Overton church: the vicar can tell us the exact height of the roof, and I will try how nearly I can approach the truth, by observing with a stop-watch how many seconds one vibration of the chandelier continues."

66

But, papa, why surely the duration of its vibration must depend upon the force which you may happen to give to the chandelier ?"

"Not in the least; and this brings us at once to the consideration of the most curious and important fact in the history of the pendulum, and for a knowledge of which we are also indebted to Galileo.* It is termed the isochronoust property, or that by which all its vibrations, whether great or small, are performed in exactly the same period of time: but that you may be better able to comprehend this subject, attend to the diagram which I have prepared for your instruction. Suppose that the swing or pendulum A B be raised to c, it will, in effect, be raised the perpendicular height E c, and in falling will describe the arc c B; and, in the point B, it will have that velocity which is acquired C by descending through c в, or by a body falling freely through the perpendicular c E. This velocity will be suf

E

B

ficient to cause it to ascend through an equal arc B D, to the same height from whence it fell at c; and since the times of ascent and descent are equal, it will describe both these arcs in exactly the same space of time. Having lost all its motion at D, it will again begin to descend by its own gravity; and in the lowest point в it will acquire the same velocity as before, which will cause it to re-ascend to c; and thus, by ascending and descending, it will perform continual vibrations in the circumference c B D; and, were it not for the resistance of the air, and the friction at the centre of motion A, the vibrations would never cease; but from these obstructions, though small, it happens that the velocity of the mass of matter at в is a little diminished in every vibration; and consequently it does not return precisely to the same points c or D, but the arcs described continually become shorter and shorter, till at length they grow insensible; and yet the very same time is required for the performance of the shorter as the longer arcs; for, although in the one case the body passes over less space, still its velocity is proportionally decreased. You perceive, then, that in an attempt to ascertain the height of a ceiling by the vibrations of a chandelier, the extent of its swing cannot alter the time which may be required for its completion. And, if you will place your little brother in the swing, you will perceive that he will return to your hand in nearly the same space of time, whether he describes a large or small arc; although this experiment must be

* This discovery was published at | may be dated the invention of the penParis, in a treatise called " L'Usage du dulum.

Cadran, ou de l'Horloge Physique Uni- + Compounded of the Greek words verselle," in the year 1639; from which loos, equal, and Xpóvos, time.

considered as extremely rude, since there are many disturbing causes for which the theory cannot possibly make any allowance. I must, moreover, warn you that, where the arc described is very considerable, the difference in the time will be greater; for in order to insure this property of vibrating through unequal arcs in equal times, it is necessary that the path of the body should describe a peculiar curve, called a cycloid (27), and not the segment of a circle; at present, however, it is not possible for us to enter into this difficult branch of science, although I trust that at some future period I shall be justified in an attempt to explain it.

"Before taking leave of the pendulum, let me state that in consequence of experiments by M. Foucalt, its vibrations have assumed a new and very unexpected interest, no less, indeed, than that of rendering palpable to your vision the rotation of the earth round its axis; and still farther by the simple adjustment of an hororary circle, over which it is made to swing, to record and exhibit to the eye the rate of its motion; and all this is seen as distinctly as we see a horse going its rounds in a mill. The explanation is equally simple :-The pendulum swinging in space will retain its parallelism of motion, and not deviate from the plane in which it began to oscillate, while the earth will revolve independent of it."

Mr Seymour having concluded his lecture, was about to return to the Lodge, when Mrs Seymour approached the party, carrying in her hands a letter, which the smile on her countenance announced to contain agreeable intelligence.

"I have just received," said Mrs Seymour, "a letter from Miss Villers, whom you must all remember as a most delightful person. I am informed that she is about to be married to the nephew of a gentleman who is at present in our neighbourhood in search of a country residence."

"Does she mention the gentleman's name?" inquired the vicar. "Mr Henry Beacham,” said Mrs Seymour.

"The nephew of Major Snapwell, I declare!" exclaimed the delighted vicar.

The whole party participated in the pleasure which their excellent friend expressed at this discovery, and Mr Seymour immediately accompanied Mr Twaddleton to Ivy Lodge, to congratulate the major, and to make such arrangements as might expedite the purchase of Osterley Park, and the consequent introduction of a family into the neighbourhood of Overton, from whose society the Seymours anticipated the highest satisfaction.

At the same time Mrs Seymour hastened to despatch a letter to Miss Villers, in order to solicit her immediate presence at Overton Lodge.

IN our last chapter we left Mr Seymour and his reverend friend on their way to Ivy Cottage: it is only necessary to state that the major received them with that satisfaction and gratitude which the nature of their visit could not fail to produce. Plans were proposed, and arrangements concluded, for the furtherance of the object we have announced: in short, in the brief space of an hour, the major had determined the course of his future life, and had framed schemes of happiness, and visions of domestic peace, which he impatiently sought to realise. The vicar was detained by the major, but Mr Seymour quitted Ivy Lodge and returned to his family. He found the children engaged at playing at marbles. Tom was displaying to his sisters many instances of his adroitness and skill in shooting at and hitting marbles.

"Why, Tom," exclaimed Mr Seymour, "how came you possessed of such a multitude of marbles?"

"By luck; good luck, papa; I won them all before the holidays; and I can assure you that my schoolfellows acknowledge me as one of the best players at ring-taw in the school."

66 Justly, then, has your merit been rewarded," said the father. "Have you not read of the skilful Roman, who could blow peas through a quill, and deposit them with such nicety on the point of a pin, placed at some distance, as rarely to miss his aim?"