Compound Forces.--The Composition and Resolution of Motion.--Rotatory Motion.--The Revolving Watch-glass.--The Sling.--The Centrifugal and Centripetal Forces.--Theory of Projectiles.--A Geological conversation between Mr. Seymour and the Vicar.

On the following morning, Mr. Seymour proceeded to explain the nature of “Compound Forces.” The young party having assembled as usual, their father commenced his lecture by reminding his auditors, that the motion of a body actuated by a single force was always in a right line, and in the direction in which it received the impulse.

“Do you mean to say, papa, that a single force can never make a body move round, or in a crooked direction; if so, how is it that my ball, or marble, will frequently run along the ground in a curved direction? Indeed, I always find it very difficult to make it go straight.”

“Depend upon it, my dear, whenever the direction of a moving body deviates from a straight line, it has been influenced by some second force.”

“Then I suppose that, whenever my marble runs in a curved line, there must be some second force to make it do so.”

“Undoubtedly; the inequality of the ground may give it a new direction; which, when combined with the original force which it received from your hand, will fully explain the irregularity of its course. It is to the consideration of such compound motion that I am now desirous of directing your attention: the subject is termed the “Composition of Forces.” Here is a block of wood, with two strings, as you may perceive, affixed to it: do you take hold of one of these strings, Louisa; and you, Tom, of the other. That is right. Now place the block at one of the corners or angles of the table: and while Tom draws it along one of its sides, do you, Louisa, at the same time, draw it along the other.”

The children obeyed their father’s directions.

“See!” said Mr. Seymour, “the block obeys neither of the strings, but picks out for itself a path which is intermediate. Can you tell me, Tom, the exact direction which it takes?”

“If we consider this table as a parallelogram, I should say, that the block described the diagonal.”

“Well said, my boy; the ablest mathematician could not have given a more correct answer. The block was actuated by two forces at the same time; and, since it could not move in two directions at once, it moved under the compound force, in a mean or diagonal direction, proportioned to the influence of the joint forces acting upon it. You will, therefore, be pleased to remember, it is a general law, that where a body is actuated by two forces at the same time, whose directions are inclined to each other, at any angle whatever, it will not obey either of them, but move along the diagonal. In determining, therefore, the course which a body will describe under the influence of two such forces, we have nothing more to do than to draw lines which show the direction and quantity of the two forces, and then to complete the parallelogram by parallel lines, and its diagonal will be the path of the body. I have here a diagram which may render the subject more intelligible. Suppose the ball B were, at the same moment, struck by two forces X and Y in the directions BA and BD. It is evident that the ball would not obey either of such forces, but would move along the oblique or diagonal line BC.”

“But,” said Tom, “why have you drawn the line BD so much longer than BA?”

“I am glad you have asked that question. Lines are intended, not only to represent the direction, but the momenta, or quantities of the forces: the line BD is, as you observe, twice as long as BA; it consequently denotes that the force Y acting in the direction BD, is twice as great as the force X acting in the direction BA. Having learned the direction which the body will take when influenced by joint forces of this kind, can you tell me the relative time which it would require for the performance of its diagonal journey?”

Tom hesitated; and Mr. Seymour relieved his embarrassment by informing him, that it would pass along the diagonal in exactly the same space of time that it would have required to traverse either of the sides of the parallelogram, had but one force been applied. Thus, the ball B would reach C in the same time that the force X would have sent it to A, or the force Y to D. “I will endeavour to prove this fact beyond all doubt. It is, I think, evident, that the force which acts in the direction BA can neither accelerate nor retard the approach of the body to the line DC, which is parallel to it; hence it will arrive at C in the same time that it would have done had no motion been communicated to it in the direction BA. In like manner, the motion in the direction BD can neither make the body approach to nor recede from AC; and it therefore follows, that, in consequence of the two motions, the body will be found both in AC and CD, and will therefore be found in C, the point of intersection.”

Louisa seemed to express by her looks the irksomeness of such demonstrations; and which did not pass unobserved.

“This may appear tedious and uninteresting,” said Mr. Seymour, “but the information is absolutely essential to our future progress: if you would reap, you must sow.”

Tom and Louisa both expressed themselves willing to receive whatever instruction their father might consider necessary; and they farther declared, that they understood the demonstration he had just offered them.

“Is it not then evident,” proceeded Mr. Seymour, “that the composition of forces must always be attended with loss of power; since the diagonal of a parallelogram can never, under any circumstances, be equal to two of its sides? and is it not also evident, that the length of the diagonal must diminish as the angles of the sides increase: so that the more acute the angle at which the forces act, the less must be the loss by composition? But I shall be better able to explain this law by a diagram. If BA, AC be the sides of a parallelogram, representing the direction of two forces, and AD the diagonal path of the body, is it not evident that the line AD will shorten as the angle BAC increases?”

“We see that at once,” cried Tom, “from the diagram before us.”

“Then we will proceed to another fact connected with the same subject. Look at this diagram; is not the diagonal AD common to both the parallelograms inscribed about it, viz. of ABCD, and AEFD?”

“To be sure it is.”

“Then it is equally clear, that a body may be made to traverse the same path AD, by any pair of forces represented by the adjacent sides of either of such parallelograms.”

“Undoubtedly.”

“I request you to keep that fact in your recollection.”

“I have now to inform you,” continued he, “that a single force may be resolved into any number of forces, and may, in fact, be regarded as compounded of innumerable oblique ones. In order, however, to render this fact more intelligible, I must refer you to fig. 6, from which it will appear that the motion of a body, along the line AD, will be the same whether it arise from one single force acting in that direction, or from two forces impressed upon it in the directions AB, AC, or in those of AE, AF; and, consequently, although the motion may, in reality, be the effect of a single force, yet it may be considered as compounded of two or more in other directions, since the very same motion would arise from such a composition.”

Tom acknowledged the truth of this statement; and Mr. Seymour assured him, that, when they came to play at ball and marbles, he should be able to give him a practical demonstration of the fact; for he would show him, that whenever a body strikes a surface obliquely, or in an inclined direction, such a resolution of force will actually take place: “and now, Tom,” said his father, “give me a marble; for I wish to explain the reason why it turns round, or revolves on its axis, as it proceeds forward.”

“I suppose,” said Tom, “it depends upon the action which I give to it by my thumb and finger when I shoot it out of my hand.”

“You are, undoubtedly, capable of thus giving to your marble a certain spinning motion, the effect of which we shall have to consider hereafter; but I fancy you would be greatly puzzled to make it proceed without revolving, give it what impulse you might by your hand.”

“I have sometimes tried,” said Tom, “to make it do so by pushing it along with a flat ruler, but it always rolled in spite of me.”

“Then it is clear, from your own experiment, that its rotation cannot arise from the cause you would assign to it. If you will attend to this diagram,” continued his father, “I will endeavour to explain the operation. It is evident that, as the marble moves along the ground BD, the motion of the point B will be retarded by the resistance occasioned by its rubbing on the ground; while the point C, which does not meet with any such resistance, is carried forward without opposition, and it consequently must move faster than the point B; but since all the parts of the marble cohere or stick together, the point C cannot move faster than B, unless the marble revolves from C to E; and as the several points of the marble which are successively applied to the floor are retarded in their motion, while the opposite points move freely, the marble during its progressive motion must continue to revolve.”

“But you said, papa, that whenever a body moved in any direction, except that of a straight line, it must have been acted upon by more than one force; and yet the marble not only runs along the ground, but turns round; at the same time, by the simple force of my hand.”

“The revolution of the marble, my dear boy, is brought about by no less than three forces: look attentively at the diagram, and you will easily comprehend my explanation. There is, in the first place, the rectilinear motion given to it by your hand; then there is the friction of the ground: since, however, this latter acts in a contrary direction, it merely tends to lessen or counteract the velocity with which the under-surface proceeds, and consequently to give a relatively-increased progressive motion to its upper part; then comes that force by which its several parts cohere, and which may be represented by CH; so that the two forces producing the revolution of the point C, are justly expressed by the lines CG, CH: but these are in the direction of the two sides of a parallelogram, the point will therefore move along the diagonal CE. I have here a toy for you, which will serve to explain still farther the causes of rotation to which I have alluded.” Mr. Seymour produced a watch-glass, in the hollow of which stood a dancing-figure of thin card, as here represented.

He placed it upon a black japanned waiter,^[16] which he held in an inclined position, when it immediately slided down the inclined plane, as might have been expected. He next let fall a drop of water upon the waiter, and placed the watch-glass in it. Under this new arrangement, instead of sliding, the watch-glass began to revolve as soon as an inclination was given to the surface; and it continued to revolve with an accelerated velocity, obeying the inclination and position of the plane, as directed by the hand of the operator.

“What a very pretty effect is produced by the rapid revolution of the figure!” observed Louisa.

“Its use in the arrangement,” said her father, “is to render the accelerated motion more obvious.”

“I perceive it revolves faster and faster, or, I suppose I ought to say, with an accelerated velocity,” said Tom.

“Certainly,” answered Mr. Seymour; “whenever a force continues to act, the motion produced by it must be accelerated for the reason already given you^[17]--but let me explain the operation of the drop of water, which, as you have just seen, converted the sliding into the revolving motion. In the first place, in consequence of the cohesion of the water to the two surfaces, a new force was introduced, by which an unequal degree of resistance was imparted to different portions of that part of the watch-glass in contact with the plane, and, consequently, in its effort to slide down, it necessarily revolved. Now, if you will attentively observe the change of figure which the drop of water undergoes during the revolution of the glass, you will perceive a species of vortex; a film of water, by capillary action, is drawn to the foremost portion of the glass, while, by the centrifugal force, a body of water is thrown under the hinder part of it; the effect of both these actions is to accelerate the rotatory motion.

“I shall now dismiss the subject for the present, but on some future occasion I shall probably revert to it; for it may be made to afford a simple illustration of the rotatory and progressive motions of the earth round the sun; and it may also give us the means of producing some optical effects of a very curious kind.”(15)

Mrs. Seymour here suggested that, as it was past one o’clock, the children should be dismissed to their more active sports in the garden.

“We will instantly proceed to the lawn,” replied Mr. Seymour, “and Tom may try his skill with the sling; an amusement which I have provided as a reward for his industry, and which will, at the same time, convey some farther information concerning the nature of those forces we have been just considering. The sling,” continued his father, as he advanced upon the lawn, “consists, as you perceive, of a leathern thong, broadest in the middle, and tapering off gradually towards both ends. To each extremity is affixed a piece of string. I shall now place a stone in the broad part of the leather, and introduce my middle finger into the loop formed in one of the strings, and hold the other extremity between my fore-finger and thumb.”

He then whirled it round, and when it had gained sufficient impetus, he let go his hold of the string, and the stone instantly shot forth with amazing velocity.

“See! see! there it goes,” exclaimed Tom; “to what a height it ascended!”

“And to what a distance has it been projected!” observed Louisa, who had attentively watched its descent.

“Now, Tom,” said his father, “can you explain the operation you have just witnessed?”

“Not exactly, papa.”

“Then attend to me. Have you not learned that circular motion is always the result of two forces?”

“Undoubtedly,” replied Tom; “of one force which attracts it to the centre around which it moves, and of another which impels it to move off in a right line.”

“Certainly; the former of these forces is, therefore, termed the centripetal, because it draws the body towards the centre, while the latter is called the centrifugal force, since its influence disposes the body to fly off from the centre. In circular motion, these two forces constantly balance each other; otherwise it is evident that the revolving body must either approach the centre or recede from it, according as the one or the other prevailed. When I whirled round the sling, I imparted a projectile force to the stone, but it was prevented from flying off in consequence of the counteracting or centripetal force of the string; but the moment I let go my hold of this, the stone flew off in a right line: having been released from confinement to the fixed or central point, it was acted upon by one force only, and motion produced by a single force is, as you have just stated, always in a right line.”

“But,” observed Louisa, “the stone did not proceed in a straight, but in a curved line: I watched its direction from the moment it left the sling till it fell to the ground.”

“You are perfectly correct,” replied Mr. Seymour, “it described a curve, which is called a parabola; but that was owing to the influence of a new force which came into play, viz. that of gravity, the effect of which I shall have to explain hereafter.”

“I cannot understand,” said Tom, “why the stone should not have fallen out of the sling when you whirled it round over your head.”

“Because, my dear, it was acted upon by the centrifugal force, which counteracted that of gravity: but I will render this fact more evident, by a very simple and beautiful experiment. I have here a wine-glass, around the rim of which I shall attach a piece of string so as to enable me to whirl it round. I will now fill it with water, and although during one part of its revolution it will be actually inverted, you will find that I shall not spill a single drop of water.”

Mr. Seymour then whirled round the glass, and the young party were delighted with the confirmation thus afforded to their father’s statement.

“I see,” said Tom, “how it happened: when the glass was inverted the water could not fall out, because it was influenced by the centrifugal force which opposed gravity.”

“Exactly. Have you ever observed what happens during the trundling of a mop? The threads which compose it fly off from the centre, but being confined to it at one end they cannot part from it: while the water which they contain being unconfined, is thrown off in right lines.”

“I have certainly observed what you state,” said Louisa; “the water flies off in all directions from the mop.”

“Yes,” added Tom, “the water was not acted upon by the centripetal force as the threads were, and consequently, there was nothing to check the centrifugal force, which carried the water off in a straight line from the centre.”

“You are not quite correct,” said Mr. Seymour; “the water does not fly off in a right line from the centre, but in a right line in the direction in which it was moving at the instant of its release; the line which a body will always describe under such circumstances, is called a tangent, because it touches the circumference of the circle, and forms a right-angle with a line drawn from that point of the circumference to the centre: but I will render this subject more intelligible by a diagram.

“Suppose a body, revolving in the circle, was liberated at a, it would fly off in the direction ab; if at c, in that of cd; and if at e, in that of ef; and so on. Now, if you draw lines from these several points to the centre of the circle, you will perceive that such lines will form, in each case, a right-angle. In the experiment which you have just witnessed, the surface of the water must have formed, during its revolution, a right-angle with the string, and consequently could not have fallen out of the wine-glass. A knowledge of this law,” continued Mr. Seymour, “will explain many appearances which, although familiar, I dare say, have never been understood by you. You may remember accompanying me to the pottery, to see the operation of the turning-lathe; it was owing to the centrifugal force produced by the rotation of the wheel, that the clay, under a gentle pressure, swelled out so regularly; from a similar cause, the flour is thrown out of the revolving mill as fast as it is ground; and I shall presently show you that you are indebted to this same force for the spinning of your top and the trundling of your hoop. But let us quit this subject for the present, and pursue the stone in its course after it is liberated from the sling. Louisa has justly observed that it described a curve; can you explain why it should deviate from a straight line?”

“Let me see,” said Tom, thoughtfully; “it would be acted upon by two forces, one carrying it forward in a right line, the other bringing it to the earth; it would, therefore, not obey either, but describe a diagonal: but why that diagonal should be a curve I cannot exactly explain.”

“Then I will give you the reason,” said his father. “A stone projected into the air is acted upon by no less than three forces; the force of projection, which is communicated to it by the hand or the sling; the resistance of the air through which it passes, and which diminishes its velocity without changing its direction; and the force of gravity, which ultimately brings it to the ground. Now, since the power of gravity and the resistance of the air will always be greater than any force of projection we can give a body, the latter must be gradually overcome, and the body brought to the ground; but the stronger the projectile force, the longer will those powers be in subduing it, and the farther will the body go before it falls. A shot fired from a cannon, for instance, will go much farther than a stone thrown from your hand. Had the two forces which acted upon the stone, viz. those of projection and gravity, both produced uniform motion, the body must certainly have descended through the diagonal; but since gravity, as you have already learned, is an accelerating force, the body is made to describe a curve instead of a straight line.

“This law, however, will require the aid of a diagram for its explanation. Let X represent the ball at its greatest altitude, XY the force of gravity drawing it downward; and XZ that of projection. We have here, then, two forces acting in the direction of the two sides of a parallelogram. In passing on to Z, the ball will perform the diagonal Xa; and in the next equal space of time, will descend through three times the distance Za, and will consequently be found at b; while in the next period it will fall through five equal spaces, and pass to c; and in the next period, again, as it must fall seven such spaces, it will reach the ground at d, having described a portion of a curve from X to d, or during the time that the two forces were in simultaneous operation. The same principle will explain the curved ascent of the ball, substituting only the laws of retarded for those of accelerated motion; for it is clear, that the body during its ascent, will be retarded in the same degree in which it was accelerated during its descent.”

“Your explanation,” said Louisa, “appears very clear and satisfactory.”

“The curve which Projectiles (that is to say, bodies projected into the air) describe, is termed a Parabola (16), although the resistance of the air, which is not recognised in the theory, produces a considerable influence on the practical result.”

The children now proceeded to amuse themselves with the sling. Louisa challenged Tom to a trial of skill. She fancied that she could hurl a stone with greater accuracy than her brother; but after several contests she acknowledged herself vanquished, for Tom had succeeded in striking the trunk of an old tree at a considerable distance, while his sister was never able to throw the stone within several yards of the mark.

“Well done, Tom!” exclaimed Mr. Seymour; “why you will soon equal in skill the ancient natives of the Balearic Islands!”

“And were they famous for this art?” asked Louisa.

“With such dexterity,” replied her father, “did they use the sling, that we are told their young children were not allowed any food by their mothers, except that which they could fling down from the beam where it was placed aloft. I fancy, however, Tom, that you would become very hungry before you could strike an object in yonder poplar.”

“At all events, I will try,” said Tom.

He accordingly whirled round his sling, and discharged stone, which flew forward with great velocity, but in a direction very wide from the mark at which it was aimed. In the next moment a violent hallooing was heard: it was from the vicar, who had narrowly escaped the boisterous salutation of the falling stone, which, in its anxiety to throw itself at the feet of the reverend gentleman, struck the beaver penthouse that defended his upper story, and by a resolution of forces which we have endeavoured to explain, darted off in the direction of the side of a parallelogram, and was thus averted from the equally sensitive antipodes of his venerable person, the brains in his head, and the corns in his shoes.

“Upon my word, young gentleman!” cried the vicar, “I expected nothing less than the fate of the giant of Gath.”

“My dear Mr. Twaddleton,” exclaimed Tom, in a tone of alarm, “I sincerely hope that you have not been struck?”

“O no! thanks to my clerical helmet, I have escaped the danger which threatened me: but, tell me, what new game is engaging your attention?”

Mr. Seymour said that he had been explaining the scientific principle of the sling, and that he hoped the vicar was prepared to afford them some information respecting its invention and history.

“The sling?” repeated the vicar; “why, bless me! I left you discoursing upon elasticity; you really stride over province after province as rapidly as if you were gifted with the seven-leagued boots of the Ogre:--but to the point in question. The art of slinging, or casting stones, is one of the highest antiquity, and was carried to a great degree of perfection amongst the Asiatic nations. It was well known and practised at a very early period in Europe; and our Saxon ancestors appear to have been very expert in the use of this missile.”

Mr. Twaddleton, being desirous of communicating history of Major Snapwell, begged that Mr. and Mrs. Seymour would allow him a few minutes’ conversation; observing that the attention of the children would be agreeably occupied during their absence by their newly-acquired amusement.

“We will then, if you please, vicar,” replied Mr. Seymour, “walk to the Geological Temple, where I have lately deposited some specimens which you have not yet seen.”

“To speak sincerely,” said the vicar, “I cannot participate in that high satisfaction which you appear to feel in collecting such hoards of broken rocks and pebbles: where can lie the utility of such labour? unless, indeed, in pursuance of your Utopian plans, you intend to Mac-adamise all the roads of science.”

“Is it nothing, my dear Mr. Twaddleton, to discover the structure of different countries?”

“Which the geologist infers,” replied the vicar, “from a few patterns, picked up at random on the road side!”

“Mr. Twaddleton,” said Mr. Seymour, “I will meet you on your own ground: you are an antiquary; if an ancient monument of art be so inestimable, is not a knowledge of the antiquity of the globe itself, at least, of equal interest?”

“I understand you: you would infer that the scriptural account of the Deluge is disproved by those Sciolists, who pretend to discover the antiquity of the globe by penetrating its caverns, with as much ease as the jockey ascertains the age of a horse by looking into its jaws.”

“You speak too flippantly of a class of philosophers who have united their efforts to investigate a sublime subject upon the true principles of science; were you to attend the meetings of the Geological Society, and hear the discussions of its members, you would cease to talk thus irreverently.”

“Although I may be unknown to your genii of the mountains, I am, at all events, acquainted with a kindred class of philosophers who rival them in industry, if not in talents; and notwithstanding the limited range of their observations--being confined to the mountainous districts they inhabit--I have little doubt but that their labours have proved as acceptable to the world as those of the disciples of Hutton or Werner. I once visited this district, and although the language of its inhabitants was entirely unknown to me, I soon discovered, by the aid of a glass, that they were in serious discourse with each other; and one of the elders of the fraternity, who was seated on a craggy precipice that overhung an extensive valley covered with rich verdure, appeared, from his gestures, as if pointing out to his fellow-labourers, who were digging in all directions in search of treasure, the danger of an approaching convulsion. While I was yet gazing, the fatal catastrophe actually occurred; immense masses of the tottering strata rolled with precipitous haste into the valley, involving in its ruin hundreds of its inhabitants. It was extraordinary to behold the effects of this shock upon those who were beyond the reach of its more destructive influence; hundreds were seen scaling heights that appeared inaccessible; others, stumbling--falling down frightful precipices--rising again--helping, or pushing each other on--the foremost serving as so many stepping-stones to those behind, who, in their turn, hauled up the clusters over whose backs they had so unceremoniously vaulted.”

“How awful!” cried Mrs. Seymour; “I never heard of any modern catastrophe of such fearful extent: where did it occur?”

“The vicar doubtless alludes to the terrible earthquake of Messina, or perhaps to that of Lisbon.”

“I neither allude to the one nor to the other,” cried Mr. Twaddleton; “and yet, in some respects, the catastrophe which I have described resembled that of Lisbon; for during the dreadful disaster human beings seen to take advantage of the confusion to murder many of the inhabitants, and to pillage their territories.”(17)

“For goodness’ sake!” cried Mrs. Seymour, “tell us at once where this terrible event occurred.”

“In a fine Cheshire cheese!” exclaimed the vicar, “which had furnished abundant food to the miniature republic of mites that occupied its deep ravines and alpine heights. I think now,” continued the reverend gentleman, “I am amply revenged for the allegorical jokes in which Mr. Seymour has so often indulged at my expense.”

“I am well satisfied,” said Mr. Seymour; “for by repeating your allegory to my children, I shall be enabled to convey a striking lesson of wisdom. They will learn from it that there is not any pursuit, however exalted, that may not be assailed by the weapons of ridicule, especially when wielded by those penurious philosophers whose ideas of utility are circumscribed within the narrow limits of direct and immediate profit.”

“It is too true,” cried Mrs. Seymour, “that we are all apt to depreciate those branches of knowledge which do not bear directly upon the comforts or necessities of life; and the applications of geology are, perhaps, so remote as scarcely to be discovered by the mass of mankind.”

“There I must differ with you,” replied her husband: “to say nothing of the practical advantages which have accrued to the miner from this study, it has been the means of bringing hundreds of acres into cultivation in districts where never a blade of grass had before grown;(18) and if scholastic researches have thrown additional light on scriptural subjects, they are no more to be compared with those of the geologist on these occasions, than is the light of the glow-worm to that of the sun.”

“Hey-day! what do I hear?” exclaimed the vicar. “Would you compare the testimony of the Apamean medal with that of an unshapen flint?”

“I would rest my faith upon a rock,” replied Mr. Seymour; “the caves of Buckland(19) have done more towards supporting the Mosaic account of the Deluge than all the medals of the virtuoso. Fossils, in truth, are to the geologist what medals are to the antiquary, preserving a record of events which must otherwise have perished in the stream of time.”

Mr. and Mrs. Seymour and the vicar by this time arrived at the Wernerian Temple, where, having discussed several points connected with its objects, Mr. Twaddleton gave an account of Major Snapwell, whose history created considerable interest, and determined Mr. Seymour to call at Ivy Cottage, and invite its inmate to the Lodge.