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has informed me of his design; .*.*. called those who love wisdom, philosophers; a man eager for glory often turns aside from the way of virtue; what slave is more attached to his master than a dog o a good man adheres to good resolutions; the age of Cicero was fruitful in virtues; the prison is filled with traffickers; a-life without friends is full of snares and fear; Italy was formerly full of Pythagoreans; Gaul abounded in men and the productions of the earth; he is without human aid; he declared that he was the third person who was destined by fate to hold supreme power in the state; it is clearly ascertained to me that you are able to gain that kingdom; beasts are devoid of reason and speech; all who are possessed of virtue are happy; anger cannot govern itself; whatever any one enjoys is his own (enjoyment is possession); Tarquin had a brother (by name) Aruns, a mán of a mild character; the fleet of Xerxes consisted of one thousand two hundred ships; he accused him of flight and fear; I free thee, O son, from blame; he is a man of no importance ; the little field was worth two thousand sesterces; your letter was of value to me; of what account is that glory of men 2 Sextilius highly valued money obtained legally; depraved men are wont to value their own property little, and to long for others’ property; it is very base to value more highly what appears useful than what appears honourable; Canius bought the gardens at the price which Pythius wished; I sell my corn for not more (pluris) than the rest, perhaps even for less; I do not care a rush for those nefarious men; let us account of no consequence the tales of ill-tempered old men; you do not value me a farthing; how much do I value you ? not a rush; this slave is worth nothing ; he allows her to play as she likes, and does not regard her at all; what is of very great importance is often considered a necessity; Julius Caesar adapted the year to the course of the sun, so that it had three hundred and sixty-five days; Claudius took very little sleep; the state of the Senons is very strong and of great weight among the Gauls; an orator ought to be a man of good judgment and of very high ability; thou also wilt become one of the celebrated fountains ; any man may err, but none except a fool will persist in error; it is the mark of a dull understanding to follow the streamlets and not to visit the fountains ; nothing is so much the token of a narrow and depraved mind as to love riches ; all these things, except the capitol and the citadel, were in the enemy's hands; you know that by this time I am altogether Pompey’s ; to put many persons in danger of their lives seems the act of a cruel man, if man he may be called; to yield to mecessity has always been considered the mark of a wise man; Popilius got the keys into his own possession; that you should come as soon as possible greatly concerns your domestic interests; your being in good health greatly concerns you and me; it is not of so much consequence in what disposition a thing is written as in what it is taken; that in no way concerns me; that concerns them more than himself; letters were devised that we might inform the absent of anything which it concerned us or (aut) them to know; your being the commander is for the common safety; it is of no consequence how many books you have, but of what kind they are ; Caesar was accustomed to say that his safety concerned not himself so much as the commonwealth ; he is sick at heart; he is very much attached to (very fond of) you; he went away with a confused and undetermined mind,

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Mr. EarToB,-Yesterday, for the first time, the PopULAR EDUCAToR was put into my hands. At page 224, I find “Garçon Embarrasse,” wishing for a solution of a certain question; but whether to clear up is own mind, or to embarrass the minds of others, we are not stricty told, . But however this may be, I would inform him that he may find it elegantly solved in Professor Davies’ “Key to Hutton's Mathematics,” page 194. e g

Mir. Editor, I perceive you have afforded a niche to arithmetic in your popular journal; permit me, therefore, as an old arithmetician, to give a few hints relating to that science. Although I am old and fast falling into the back-ground before my youthful, more highly gifted, more highly favoured, and rapidly improving countrymén, yet my arithmetic, as yet, is in tolerable repute, because perhaps it is original. I was a very poor boy, brought up in the most servija drudgery at the plough; no school, no teacher, I had to glean for knowledge by the way-side, and that, too, then (more so than now; in a very barren land. But by some kind instingt I knew the principles of the fundamental rules in arithmetic before I knew their fiames. As years came on, and I began to see the work of others, I became a critic. A few of my pertinent remarks on teaghers and writers on arithmetic I will lay before you, I quarrelled first with the word irust either éxpressed or implied.

*In subtraction you must when you borrow i, carry ten to the next |

figure in the lower line;” whereas I, in my original conception, when I borrowed 1 (as old men and old writers call it), I reduced the figure in the upper line by unity, and this I found on mature reasoning to be philosophically correct. Mr. Augustus De Morgan in his “Principles of Arithmetic "gives a reason for the popular method, but it seems to want originality. “To put the less line under the greater” should be discarded altogether. “In multiplication, you nust place the multiplier wmder the multiplicand, you must begin with the unit figure of the multiplier,” &c. I will now work an example in three different ways, in which every one of these rulee is violated.

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I do not mean to say that every example in multiplication is so well adapted for contractions as the one I have chosen, but I do say that many are so. In the first method, it may be observed that the sum of the 1st and 2nd, as well as of the 4th and 5th partial products of the multiplicand, is equal to the middle. In the second method, the first partial product is that of 9 times the multiplicand, for 900; the second partial product is that of 5 times the former, for 45;000; and the third partial product is that of three times the first, for 27; hence, the total product is that of 45,000, god, 27–15927. In the third method, the first partial product is that of 12 times the multiplier, for 1,200; the second is that of times the former, for 96; the third is that of 7 times the second for 672,0000; hence, the total product is that of 672,000+1,200--96= 6721296.

In division, we are told that “the product of the divisor by the quotient figure must be less, or at least not more, than the dividend;” even this is not absolutely necessary, although I admit that it is, generally speaking, the most convenient. But in the finding of the greatest common measure it may frequently be dispensed with to great advantage. The olden writers seem to insist that in the Rule of Three the first and third terms must be of the same kind; but the moderns, upon whom new light has broken in, all say, without doubt correctly, that “the first and second must be of the same kind.” With me it has been always a matter of indifference, and in a strict numerical sense there is really no difference. Another hindrance to the young tyro is the mysterious language in which the rules of arithmetic are enveloped. Every new enunciation of the application of its fundamental principles is dignified with the appellation of a new rule; hence, the question commonly put: “How far can you cipher P’” “What rule are you now in P’’ For my own part, I never could discover but two rules, Addition and Subtraction; or, for the sake of conformity, if you please, four. Our old mathematicians liked to be accounted conjurers, and the present ones evince a desire to mystify their knowledge, or to let it out as slowly as possible. Hence, youths are suffered to believe that they are about to learn a certain species of magic, that figures are tools selected for this purpose, and that a configuration of these tools, according to unexplained rules, gives them the mastery over secrets otherwise unattainable. They are left to think that to add whole numbers, and to add vulgar fractions, are different operations; and thuss they stumble at every step. There is no man, however illiterate, but he can add and subtract; from this, therefore, it is evident that arithmetic is the easiest of all the sciences. Why, then, should we embarrass a science founded on common sense with so many unnecessary difficulties? Yours, &c.

TRostas Gus DRY.

Huel Wor, near Helsion, Gornwall, Sep. 19, 1853.

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Z-T N SIR,--I beg leave to send you a sketch of a balance which I have designed, with a view to bring such an instrument within the reach of any student of chemistry; the expense to an ingenious person would scarcely exceed a shilling. The balance is composed of a enknife blade, easily procurable at any cutler's for a few pence, xed in a stand of wood; the beam, made of a bit of polished brass wire, is formed to rest on the edge of the blade as shewn in the sketch; and the pans are watch glasses borne by silk threads; on the top of the loop in the beam, I have soldered an index which can be adjusted (the balance being at perfect rest) by a card with a zero point drawn on it, sliding in a groove on the top of the wooden stand. This balance will be found inexpensive, and sufficiently sensitive for all usual purposes, weighing grains with accuracy. I am, &c., John J. SLOANE. 114, Great Britain-street, Dublin. Sep. 17, 1853.

S O L UTION S.

*olution of the Question proposed in No. 75, page 344, Vol. III., of the “Popular Educator.”

Here, the wolf would eat # of the sheep in 20 minutes, and the tiger would eat # of the sheep in 10 minutes; therefore, both would eat # of the sheep in 20 minutes. Consequently there would be # of the sheep to be eaten together by the wolf, tiger, and lion.

Now, the wolf would eat 1 sheep in 1 hour; the tiger would eat 3 sheep in 1 hour; and the lion would eat 2 sheep in 1 hour; therefore, all would eat 6 sheep in 1 hour. In what time, then, would they eat # of a sheep Here we have 6 sheep : # sheep :: 1 hour ; 13 minutes for the time taken by the 3 animals to eat the whole sheep, from the commencement of the operation.

Lastly, we have 60 minutes : 21% minutes : : 1 sheep:#, the part the wolfate; 20 minutes : 11% minutes: : 1 sheep : ##, the part the tiger ate; 30 minutes: lä minutes :: 1 sheep: #3, the part the lion ate. Whence, #-F#-F#-šā-1 sheep, proof.

G. ARCHBoLD, St. Peter's.

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Thirteen thirty-sixths of the whole, the Wolf finish'd,
Twenty-one thirty-sixths, the Tiger diminished,
Leaving only two thirty-sixths, full to the King,
The portion eaten up, by himself, at a spring.
Having found out these answers, apart from all men,
I beg to subscribe myself, A. U., Rutherglen.

[This question was also correctly solved by J. W., Reading; W Parker, Busk; H. R. R. ; Josephus, Gravesend; R. Brown, Leven H. C. P., Bristol; and others.]

ANSWERS TO CORRESPONDENTS.

G. A., jun. (Liverpool): We don’t know the French dictionary he speaks of. Cassell's French Dictionary has the pronunciation wherever there is any difficulty.—S. E. (Sheffield): We think our own system the best. G. M. (Aldersgate) informs us that quilled glass may be easily had under the name of tube glass, and that it may be &: ased at Mr. Gibbon's glass and bottle warehouse, Jerusalem-passage, Clerkenwell, where spirit-lamps and other chemical apparatus may also be had. A SUBSCRIBER (Bromley) recommends such of our subscribers, as can spare the money to do theimselves the pleasure of presenting vol. i. of the P. E. to some promising, aspiring young friend He seems to think that by this means many would be greatly encouraged in the work of †: ment, and at the same time be induced to buy the whole series. He has tried the plan himself, and expects good results.—PHILO (Nottingham): The instruments he mentions will be described in future numbers. ALPHA (Thornton Le Clay): The term instinct is generally applied to the reasoning power of the lower animals, and is considered to be an instantaneous faculty of judging of what is right and wrong as regards their welfare, conferred on them by God. The term reason is applied to that faculty with which God has endowed man, to enable him to judge of what is right and wrong as regards his welfare; conscience is no other than this faculty properly instructed, or made aware of what is right and wrong, and of their consequences. Hence, the man who has received the highest moral and religious educationn has the most tender conscience. When the conscience is seared by neglecting its warnings, the possessor of it becomes worse than the most ignorant savage, and in his actions falls lower than the brute creation. We recommend all our readers, by all means, to cultivate a tender conscience, and one void of offence towards God and man. The penny edition of thc P. E. Huay be exchanged for the three-halfpenny edition, if it be quite clean and in good condition, on paying the difference in price. ToM HARRIson (Greenwich): We shall discuss the subject of the Bino-: mial Theorem in the lessons in Algebra as soon as we can-J. S. (Dartford) Go on improving.—F. RICHARDs (Selby): We don’t know it.—W. R. EDWARDs (Swinton-st.): Thanks—B. J.R.: We quite agree with his remarks on Prop. III. Book I., but we have dwelt already too long on the initial propositions; we must now advance with more speed. OMEGA : We think that the Latin Dictionary by E. A. Andrews, which is a translation of Dr. Freund's Latin Dictionary published in Germany, is most likely to be the best. As to the study of Latin, get all the knowledge you can by hook or by crook.-J. E. S. A. is too flattering to us; we shall consider his suggestions. J. E. (Oldham), STUDENT IN FRENCH (Leeds): Yes.—JAMEs Jones (Morriston): Apply, and go ahead, – ConstanT ADMIRER. (Torquay) deserves our sincerest thanks; but many men, many minds.-WARIN (East Dereham): Ah! my friend, beware 1 there are sad flaws in Legendre's Geometry; don’t forsake old Euclid; he has stood 2,000 years l The second case of Prop. VII. won't do, neither will the demonstration of Prop. XVI.-JENobiA (Brighton): The “Historical Educator” is a substitute for the : in History in the P. E.-J. THoMPson (Leicester): We are not Certallie J. F. ENTwistLE (Wigan): His tables for the Octary Scale are very ingenious; many thanks for his kind endeavours on our behalf.-E. HART: For a list of French books, write to any of the foreign booksellers in London, as D. Nutt, Strand; Dulan and Co., Soho-square, &c. The best library in London for scientific and all other books is that of the British Museum; admission is free, but you must have a recommendatory letter from some gentleman who is well known, addressed to the chief librarian,—A SUBsch IBER (Westminster) need be under no alarm about omissions of sections in any branch; misprints will sometimes happen.—W. G. R. VENNER had better make very considerable progress in learning before he thinks of the ministry; mere spouting won’t do.

L IT E R A R Y NOTIC E S.

GREEK.

The Third Volume of CAssell's CLAssical, LIBRARY will contain the Acts of the Apostles in the original Greek, according to the text of Augustus Hahn; with foot. historical, and expository Notes ; followed by a Lexicon, explaining the meaning of every word-the whole carefully revised and corrected. This work is well adapted for the use of Schools, Colleges, and Theological Seminaries, and will supply our Greek students with excellent materials for practice in translation.

LATIN.

The first volume of CAssell's CLASSIgAL LIBRARY is now ready, price ls. 6d., containing. Latin extracts for translation on the following subjects—Easy Fables, Mythology, Biography, The History of Rome, and Ancient Geography; with a suitable Dictionary. The second volume, which is publishing in weekly... numbers price , 2d. each, will consist of useful Latin Exercises, or English sentences, to be translated into Latin, with numerous references to Andrews and Stoddart's Latin Grammar, a F valuable treatise now in the press, - or

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ON PHYSICS OR NATURAL PHILosophy. No. V.

LAWS OF FALLING BODIES, INTENSITY OF GRAVITY, &c.

Falling Bodies.—The three laws of Falling Bodies are the following, which are only strictly accurate when the consideration of the resistance of the air is omitted; or, in other words, when the bodies fall in a perfect vacuum. 1st Law: All bodies, large or small, fall with equal rapidity to the earth's surface, at the same place. This law is proved by the following experiment, called the guinea and feather experiment. Take a tube of glass of about two yards in length, and a convenient diameter, fig, 11, closed at one of its extremities, and furnished at the other with a brass stop-cock; put into this tube, placed vertically, with the closed end lowest, any two bodies of different densities, such as lead and cork, gold and paper, &c., and make a vacuum in it with Fig. 11. an air-pump ; then quickly invert the Zo tube, by placing the closed end uppermost, #| || and keep it in the vertical position; you T will now see the light body and the heavy body, such as the guinea and the feather, both fall to the other end of the tube with the same velocity. Readmit a little air by opening the stop-cock, invert the tube in the same manner as before, and you will see the light body falling more slowly than the heavy one, in proportion to its comparative weight. Lastly, readmit the air completely, perform the same inversion, and you will find that the light body falls still more slowly than before, in consequence of the greater effect of the resistance of the air when fully admitted into the tube. The conclusion | from these experiments is, that if in the ordinary circumstances of the atmosphere | - bodies fall to the ground with unequal velocity, the cause of this is the resistance | of the air, which is more sensibly observed on the lighter bodies, and not from any i difference in the action of gravity upon different substances, for it acts alike upon all substances, making them fall from the same height in the same time in a vacuum. Moreover, under equal volume, all bodies } experience the same resistance of the air in §d falling ; but the force with which they are T attracted overcomes this resistance in proportion to their mass. The resistance of the air to falling bodies is particularly evident in the case ; of liquids. When they fall in the air, they separate and fall in drops; but when they fall in a vacuum, they fall like a solid mass, without separating into drops. This phenomena is proved by the apparatus called the water-hammer; this is a tube o:- of glass of about an inch in diameter, s and about a foot or sixteen inches long, nearly half filled with water and hermetically sealed, after the air has been expelled by raising the water to the boiling point. When this tube is quickly inverted, the water in falling strikes against its lower end with a smart dry sound like that of the collision of two solid bodies. 2nd Law. The velocity acquired by a body falling in a vacuum is proportional to the time of falling. Thus, at the end of 2, 3, 4, &c., times a given unit of time, the velocity acquired will be 2, 3, 4, &c., times the velocity acquired in that unit. 3rd Law. The spaces described by a body falling in a vacuum, are proportional to the squares of the times of falling. Thus, if the times of falling be 1, 2, 3, 4, 5, &c., times a given unit of time, the spaces described will be 1, 4, 9, 16, 25, &c. times the space described in that unit. g Since, according to the third law, the space deseribed in the

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1st unit of time is 1, and the spaces described in 2, 3, 4, 5, &c., units of time, are 4, 9, 16, 25, &c., it follows that the space described in the 2nd unit of time is 4 less l, that is, 3; in the 3rd unit it is 9 less 4, that is, 5; in the 4th unit, 16 less 9, that is, 7; and so on. Hence, the spaces described in the 1st, 2nd, 3rd, 4th, &c., units of time are successively 1, 3, 5, 7, &c., according to the series of odd numbers. From this it is evident that the spaces described increase by equal quantities in equal times, which is in accordance with the definition already given of uniformly accelerated motion. The laws of falling bodies are only true when the bodies fall in a vacuum, and from heights in the atmosphere differing little from each other in comparison with the radius of the earth. When the bodies fall in the air, these laws are modified by the resistance of the atmosphere; and when they fall from very unequal heights in the atmosphere, the force of gravity is not strictly the same. Galileo, an Italian philosopher and Florentine nobleman. was the first who made the discovery of these laws, and announced them to the students of the university of Pisa, where he was professor of the mathematics in 1611 A.D.

Inclined Plane.—Various apparatus have been invented for the purpose of proving the laws of falling bodies; Galileo employed the inclined plane in an original manner; Atwood invented the machine known by his name; and M. Morin, director of the “Conservatoire des Arts and Metiers” at Páris, constructed an apparatus first proposed by M. Poncelet.

An inclined plane is one which makes with a horizontal plane any angle less than a right angle. In proportion to the smallness of the angle between these planes, so is the decrease of the velocity of a body which descends along the inclined plane. Thus, let A B, fig. 12, represent an inclined plane, A c the horizontal plane, and P C a perpendicular to the horizontal

Fig, 12.

As

plane drawn from any point B in the inclined plane. If any body M rest upon this inclined plane, its weight P acting vertically may be resolved into two forces Q and F, the one acting in a perpendicular and the other in a parallel direction to the inclined plane A c. The first force, Q, will be completely counteracted by the resistance of the inclined plane which acts in the direction Q G, and the other force F only will act on the mass of the body M in order to make it descend along the plane.

In order to ascertain the value of the force F, take on the line GP a length the number of whose units represents the weight P, and complete the parallelogram D G E H ; then the force F will be represented by the number of units of length in G D. But the triangles D G H and A B C are similar, because their angles are equal (Cassell's Euclid, Book VI., Prop. IV.); whence we have

G. H. : D G : : A B : B C, Or
P : F :: A B : B c;

that is, the force F will be less than the weight P, in proportion as the height B C of the inclined plane is less than its length. A c. Thus we can make the force F as small as we please, by diminishing the height of the plane, or the angle which it makes with the horizon, and thus slacken the motion of the moveable body M, so as to be able to take account of the spaces described in one, two, three, &c., seconds, and this without altering the laws of the motion, since the force F is continued or constant. By such experiments as this, Galileo discovered that the spaces described increased as the squares of the times.

Atwood's Asachine.—The laws of falling bodies, however, were more clearly demonstrated experimentally by means of a machine invented by Mr. Atwood, professor of chemistry in the University of Cambridge. This machine is formed of a

narrow wooden pillar, about seven and a-half feet high,

83.

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having on the top a glass case, in which is placed a brass pulley it, fig. 13. Over this pulleypasses a silken thread, so fine that its weight need not be taken into account, and having two equal weights M and M' suspended at its extremities. The axle of the pulley, instead of resting on two fixed bearings, is supported on the circumferences of four moveable wheels. By this arrangement the axle of the pulley transmits its motion to the four wheels, and the sliding friction of fixed bearings is converted into the rolling frietion of the wheels, a contrivance by which the friction of the axle is very much diminished.

On the pillar is fixed a clock-movement H, which regulates a seednds' pendulum P by means of an anchor escapement. This escapement is shown on the dial-plate above the swing-wheel which occupies the centre. This escapement oscillates with the pendulum, and inclining to the right and left alternately at each oscillation, it allows one tooth of the swing-wheel to escape. The axis of this wheel carries at its anterior extremity an index marking seconds, and at its posterior extremity, behind she dial-plate, an eccentric, shown at E on the left of the pillar. This eccentric moves with the index, and presses on a lever D, which, by its motion, overturns a small platform 1, employed to support the mass Ms.

Parallel to the pillar, and fastened to its base, is a wooden scale of nearly the same length, divided into inches and tenths of an inch, used for the purpose of measuring the spaces described by the falling body. On this scale are two stages A and B, which by means of tangent screws can be adjusted to any required height. The stage A is intended to receive the weight M! at the end of its course; and the stage B, which is hollow, allows this weight to pass through it, and is used only to stop the progress of the additional weight m which rests upon it at starting. The use of Atwood's machine is to diminish the velocity of a falling body, and to produce at pleasure a uniform motion, or a motion uniformly accelerated.

In order to understand the nature of this machine, suppose that a small piece of brass m, which in the engraving rests on the stage B, falls alone; let its velocity at the end of a second be denoted by g; its momentum or quantity of motion will then be denoted by ong. If this piece of brass m be placed on the weight M', when at the top of the scale, it will descend and gommunicate part of its motion to the two weights M and M'; for previously to this, the two weights being equal were in equilibrium, the action of gravity in each being mutually balanced. Tt is plain that the same force which would cause

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9 33 be only one thirty-third part of the velocity which it would have if it fell freely in the air. By this means we can more easily ascertain the nature of the force which causes bodies to fall, and also render the resistance of the air imperceptible. The first experiment performed by this machine proves that the spaces described by a falling body increase as the squares of the times. The pendulum P being at rest, and the second index being beyond zero, the weight M' is placed on the platform I, and is loaded with the additional weight on, the whole being k; in the horizontal position by the extremity of the lever D, and corresponding to zero on the scale. Removing then the hollow stage B, and preserving only the final stage A, place the latter, by trials, at such a distance from the zero point at 1, that from this point to the stage. A the weights in and M take only one second in falling, the fall commencing at the instant when the pendulum having been put in motion the index reaches zero on the dial-plate; for at this point the lever D is put in motion by the eccentric, and the platform I is overturned, setting the weights on and M at liberty to fall. Suppose now that we have found the height of the fall, or the space descended in a second, to be seven divisions of the scale; then repeat the experiment as before, but remove the stage A to a distance from the zero point I, equal to four times the preceding distance, that is, to the 28th division of the scale, and it will be seen that this space is described in exactly two seconds by the two weights on and M'. In like manmer it will be found that at a distance nine times the first, or at the 63rd division of the scale, the space will be described in three seconds; and so on. The third law is, therefore, verified by experiment. In order to verify the second law by experiment, it must be recollected that in accelerated motion the velocity at a given instant is that of the uniform motion which follows upon the accelerated motion. Hence, in order to discover according to what law the velocity of a falling body varies, we have only to measure the velocity of the uniform motions which immediately follow the accelerated motions successively after one,

should have a = or ; that is, the velocity of the mass would

two, three, &c., seconds of the fall.

The determination of the uniform motion after the accelerated motion is obtained by means of the stage B. This is placed just at the distance from the zero of the scale which the two weights on and M' when descending reached in a second, as in the first experiment; then, the additional weight on being stopped in its descent by the stage B, the weight M' continues to descend alone, until it be stopped by the stage A, which is placed below B at such a distance as that the weight M' shall occupy only one second in passing from B to A. Now, from I to B the motion is uniformly accelerated, and from B to A it is uniform; for the weight on being stopped by the stage B, gravity no longer acts from B to A, and the motion is only continued in consequence of the inertia of the weight M'. The number of the divisions of the scale passed over by the weight M' from the one stage to the other will then represent the velocity acquired by the two weights m and M' at the end of one second.

In repeating this experiment, the stage B is lowered to such a distance that the two weights on and M' take two seconds to descend from the point I to the stage B; the stage A is then lowered to a distance from B double of that at which it was in the first experiment. Thus, the two weights fall during two seconds in a state of uniformly accelerated motion; then reaching the stage B, the weight M' alone passes over the interval between the stage B and the stage A. The velocity acquired at the end of two seconds is therefore double of that acquired

at the end of one second. Similar experiments being made for three, four, &c., seconds, it will be found that the velocities acquired are three, four, &c., times the velocity acquired at the end of the first second ; and thus the second law is verified. M. Morin's Apparatos. – In this apparatus, or continued indicator of motion, the uniform rotatory motion of a cylinder covered with paper is combined with the motion of a falling body, in such a manner that by means of a pencil properly adjusted for the purpose, it describes on the paper a curve which represents the law of the motion. In fig. 14, the cylinder A, which is covered with paper, is about 9% feet in height, and about 16 inches in diameter; this cylinder is set in motion by a weight P, and this motion is communicated by means of a cord to the drum B ; this drum, by means of two bevelled

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wheels, communicates the motion to a rod H and to a wheel

and pinion I and o, which put the cylinder A in motion. The weight P having a tendency to accelerate its motion

during its descent, M. Wagner, the maker of the apparatus, em

:

to guide a long wooden ruler which is applied to the cylinder and is used to trace on its surface two kinds of equidistant lines, the one in planes perpendicular to the axis of the cylinder, and the other vertical. The cast-iron pieee, or monkey, M., guided in its descent by two straight iron wires, F and G, firmly fixed at their extremities, is placed at first in a catch at D, which can be opened at pleasure by drawing the wire L. To this monkey M is fastened at It the pencil which describes, during its descent, the curve S R on the cylinder as it revolves. From the forma of this curve the laws of motion are deduced. For the space passed over by the pencil at the end of any given time, is at the point on of the curve equal to the portion a sm of the vertical traced on the surface of the cylinder. But the motion of the cylinder being aniform, we can take for the duration of the faii, when the moveable has descended to in, the are h ot, between the point a, ano; the vertical which is

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ployed, for the purpose of regulating the motion of the drum is, a regulator of which the mechanism is concealed in the igure. It is known in mechanics, however, by the name of the differential motion, and it depends both on the motion of a pendulum o, and of a fly furnished with leaves, which moves with great rapidity. This fly is contained in a drum T, which rises or falls according to the velocity of the apparatus. When the motion is accelerated and the pendulum oscillates too rapidly, the drum rises, and the leaves of the fly then meeting with the resistance of the air, the motion is retarded. On the other hand, when the velocity diminishes, the drum is lowered, and the fly then meeting with less resistance from the air, the motion is accelerated. Thus a motion sensibly uniform is obtained ; and for this purpose the descent of the weight P for about 20 inches is sufficient.

The wheel N, fixed on the axis of the cylinder, is employed

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Kittiw in through the point at the origin, or the beginning of the motion of the pencil. in like manner, at any other point m, of the curve, the space passed over is represented by a' in" and the time by h' sn’. Now, by comparing the lengths a on and a' m' with the arcs h m and h' mis, we find that the lengths or distances a wi and & on' are to one another as the squares of their corresponding arcs; thus it is clearly demonstrated that the spaces passed over are to one another as the squares of the times of passing over; and we therefore conclude that the motion of falling bodies is one uniformly accelerated. The ratio which is found to subsist between the arcs h m, h'm', &c., and the verticals a m, a go, &c., show that the curve sR is a parabola whose axis is parallel to the generatrix of the cylinder; and this is at once demonstrated by unfolding on a plane the paper cover of the cylinder on which the curve is

traced by the pencil,

[graphic]
[graphic]
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