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has informed me of his design; Pythagoras called those who love figure in the lower line;" whereas I, in my original conception, wisdom, philosophers; a man eager for glory often turns aside when I borrowed 1 (as old men and old writers call it), I reduced the from the way of virtue; what slave is more attached to his master figure in the upper line by unity, and this I found on mature reasonthan a dog a good man adheres to good resolutions; the age of ing to be philosophically correct. Mr. Augustus De Morgan in his Cicero was fruitful in virtues; the prison is filled with traffickers; “Principles of Arithmetic "gives a reason for the popular method, a life without friends is full of snares and fear; Italy was formerly but it seems to war originality. To put the less line under the full of Pythagoreans; Gaul abounded in men and the productions greater” should be discarded altogether. “In multiplication, you of the earth; he is without human aid ; he declared that he was j must place the multiplier under the multiplicand, you must begin the third person who was destined by fate to hold supreme power with the unit figure of the multiplier,” &c. I will now work an in the state; it is clearly ascertained to me that you are able to example in three different ways, in which every one of these rules gain that kingdom; beasts are devoid of reason and speech; all is violated. who are possessed of virtue are happy, anger cannot govern itself; whatever any one enjoys is his own (enjoyment is possession);

1st method.

2nd method. Tarquin had a brother (by name) Araps, a mún of a mild character; the feet of Xerxes consisted of one thousand two hundred ships;

6721296

6721296 he accused him of flight and fear; I free thee, O son, from

45927

45927 blanie; he is a man of no importance; the little field was worth two thousand sesterces; your letter was of value to me; of what

26885184

60491664 account is that glory of men ? Sextilius highly valued money

33606480

302458320 obtained legally; depraved men are wont to value their own

60491664

181474992 property little, and to long for others' property; it is very base to

13442592

308688961392 value more highly what appears useful than what appears honour

47049072 able; Canius bought the gardens at the price which Pythius wished; I sell my corn for not more (pluris) than the rest, perhaps

308688961392 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;

3r3 method, you do not value me a farthing; how much do I value you ? not á rush; this slave is worth nothing; he allows her to play as she

6721296 likes, and does not regard her at all; what is of very great import

45927 ance is often considered a necessity; Julius Caesar adapted the

551124 year to the course of the sun, so that it had three hundred and sixty-five days; Claudius toók very little sleep; the state of the

4408992 Senons is very strong and of great weight among the Gauls; an

30862944 orator ought to be a man of good judgment and of very high ability;

308688961392 thou also wilt become one of the celebrated fountains; any man may err, but noné except a fool will persist in error; it is the mark of a dull understanding to follow the streamlets and not to visit well adapted for contractions as the one I have chosen, but I do

I do not mean to say that every example in multiplication is so the fountains depraved mind as to love riches ; all these things, except the capitol say that many are so. In the first method, it may be observed and the citadel, were in the enemy's hands; you know that by that the sum of the 1st and 2nd, as well as of the 4th and 6th this time I am altogether Pompey's; to put many persons in partial products of the multiplicand, is equal 10 the middle. In the danger of their lives seems the act of a cruel man, if ınan he may second method, the first partial product is that of 9 times the mulbe called; to yield to_nnecessity has always been considered the tiplicand, for 900; the second partial product is that of 5 times the mark of a wise man ; Popilius got the keys into his own possession; times the first, for’27; hence, the total product is that of 45,000

product is that of three that you should come as soon as possible greatly concerns your times the first, for

27; hence, the total product is that of 45,000 domestic interests; your being in good health greatly concerns that of 12 times the multiplier, for 1,200; the second is that of

is

8 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 672,0000; hence, the total product is that of 672,000+1,200+96-

times the former, for 96; the third is that of 7 times the second for

6721296. 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 In division, we are told that “the product of the divisor by the safety;

it is of no consequence how many books you hare, but of quotient figure must be less, or at least not more, than the diviwhat kind they are ; Caesar was accustomed to say that his safety dend;" even this is not absolutely necessary, although I admit that concerned not himselt so much as the commonwealth ; he is sick it is, generally speaking, the most convenient. But in the finding of at beart; he is very much attached to (very fond of) you; he went the greatest common measure it may frequently be dispensed with away with a confused and undetermined mind.

to great advantage. The oiden 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, with

out doubt correctly, that "the irst and second must be of the same CORRESPONDENCE.

kind." With me it has been always a matter of indifference, ånd

in a strict numerical sense there is really no difference. Another ARITHMETIC.

hindrance to the young tyro is the mysterious language in which

the rules of arithmetic are enveloped. Every new enunciation of MR. EDITOR,—Yesterday, for the first time, the POPULAR EDUCA- the application of its fundamental principles is dignified with the TOR was put into my hands. At page 224, I find " Garçon Embar- appellation of a new rule; hence, the question commonly put: rasse,” wishing for a solution of a certain question ; but whether to

" How far can you cipher?" "Whatrule are you now in ?" For clear up tis own mind, or to embarrass the minds of others, we are my own part, I never could discover but two rules, Addition and not stric:y told. But however this may be, I would inform hina Subtraction; or, for the sake of conformity, if you please, jour. that he may find it elegantly solved in Professor Davies' " Key to Our old mathematicians liked to be accounted conjurers, and the Hutton's Mathematics,” page 194.

present ones evince a desire to mystify their knowledge, or to let it Nii. Editor, I perceive you have afforded a niche to arithmetic out as slowly as possible. Hence, youths are suffered to believe in your popular journal; permit me, therefore, as an old arithme- that they are about to learn a certain species of magic, that figures tician, to give a few hints relating to that science. Although I am

are tools selected for this purpose, and that a configuration of these old and fast falling into the back-ground before my youthful, more tools, according to unexplained rules, gives them the mastery over highly gifted, more highly favoured, and rapidly improving country- secrets otherwise unattainable. They are left to think that to add men, yet ny arithmetic, as yet, is in tolerable repute, because whole numbers, and to add vulgar fractions, are different operaperhaps it is original. I was a very poor boy, brought up in the tions; and thus they stumble at every step. There is no man, most servija drudgery at the plough; no school, no teacher, I had however illiterate, but he can add and subtract; from this, thereto glean fer knowledge by the way-side, and that, too, then (more so fore, it is evident that arithmetic is the easiest of all the sciences. than nor; in a very barren land. But by some kind instinct I Why, then, should we embarrass a science founded on common knew the principles of the fundamental rules in arithmetic before seuse with so many unnecessàry difficulties? Youřs, &c. I knew their names. As years came on, and I began to see the work of others, I becanié à critic. A few of my pertinent remarks

THOMAS GUNDRY. on teachers and writers on arithmetic I will lay before you. I quarrelled brst with the word ŠUŠT either expressed or implied. Huel Vor, near Heiston, Cornwall, "Ia subtraction you must when you borrow 1, catiỹ ten to tke next

Sep. 19, 1863.

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SLOANE'S BALANCE.

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.

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[This question was also correctly solved by J. W., Reading; W Parker, Busk; H.R.R. ; Josephus, Gravesend ; R. Brown, Leren 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 purchased 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 theinselves 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 Self-Improvcment, and at the same time be induced to buy the whole serles. 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): T'he term instinct is generally applied to the reasoning power of the lower animals, and is considered to be an instantaneous faculty of judging of wbat 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 SIR, I beg leave to send you a sketch of a balance which I have instructed, or made aware of what is right and wrong, and of their consedesigned, with a view to bring such an instrument within the reach educatinn has the most tender conscience. When the conscience is seared by

quences. Hence, the man who has received the highest moral and religious of any student of chemistry; the expense to an ingenious person neglecting its warnings, the possessor of it becomes worse than the most would scarcely exceed a shilling. The balance is composed of a ignorant savage, and in his actions falls lower than the brute creation. We penknife blade, easily procurable at any cutler's for a few pence, recommend all our readers, by all means, to cultivate a tender conscience, ixed in a stand of wood; the beam, made of a bit of polished brass and one void of offence towards God and man. The penny edition of the wire, is formed to rest on the edge of the blade as shewn in the and in good condition, on paying the difference in price.

P. E. may be exchanged for the three-halfpenny edition, if it be quite clean sketch; and the pans are watch glasses borne by silk threads; on the TOM HARRISON (Greenwich): We shall discuss the subject of the Bino.: top of the loop in the beam, I have soldered an index which can mial Theorem in the lessons in Algebra as soon as we can.-J. S. (Dartford) be adjusted (the balance being at perfect rest) by a card with a Go on improving. F. RICHARDS (Selby): We don't know it.-W. K. zero point drawn on it, sliding in a groove on the top of the wooden EDWARDS (Swinton-st.): Thanks-L.J.R.: We quite agree with his remarks stand.

This balance will be found inexpensive, and sufficiently on Prop. III. Book I., but we have dwelt already too long on the initial prosensitive for all usual purposes, weighing grains with accuracy. 1 positions; we must now advance with more speed.

OMEGA: We think that the Latin Dictionary by E. A. Andrews, which is am, &c.,

JOHN J. SLOANE. a translation of Dr. Freund's Latin Dictionary published in Germany, is 114, Great Britain-street, Dublin.

most likely to be the best. As to the study of Latin, get all the knowledge Sep. 17, 1853.

you can by hook or by crook.-J. E. S. A. is too flattering to ins; we shall consider his suggestions.

J. E. (Oldbam), STUDENT IN FRENCH (Leeds): Yes.-JAMES Jones

(Morriston): Apply and go ahead, - CONSTANT ADDIRER (Torquay) SOLUTIONS.

deserves our sincerest thanks; but many men, many minus.-WARIN (East Dereham): Ah! my friend, beware! there are sad flaws in Legendre's Geo

metry ; don't forsake old Euclid; he has stood 2,000 years! The second case solution of the Question proposed in No. 75, page 344, Vol. III., of of Prop. VII. won't do, neither will the demonstration of Prop. XVI. the Popular Educator.

JENOBIA (Brighton): The "Historical Educator" is a substitute for the

lessons in History in the P. E.-J. THOMPSON (Leicester): We are not Here, the wolf would eat of the sheep in 20 minutes, and certain. the tiger would eat of the sheep in 10 minutes; therefore, ingenious; many thanks for his kind endeavours on our behalf.-E. Hart:

J. F. ENTWISTLE (Wigan): His tables for the Octary Scale are very both would eat of the sheep in 20 minutes. Consequently For a list of French books,

write to any of the foreign booksellers in London, there would be of the sheep to be eaten together by the wolf, as D. Nutt, Strand; Dulan and Co., Boho-square, &c. The best library in

London for scientific and all other books is that of the British Museum; tiger, and lion.

admission is free, but you must have a recommendatory letter from soine Now, the wolf would eat 1 sheep in 1 hour; the tiger would gentleman who is well known, addressed to the chief librarian.--A SUBSCHIeat 3 sheep in 1 hour; and the lion would eat 2 sheep in 1 BER (Westminster) need be under no alarm about omissions of sections in any hour; therefore, all would eat 6 sheep in 1 hour. In what branch; misprints will sometimes happen.-W.G. R. VENNER had better

make very considerable progress in learning before he thinks of the minis. time, then, would they eat $ of a sheep? Here we have try; mere spouting won't do. 6 sheep : sheep :: 1 hour : 14 minutes for the time taken by the 3 animals to eat the whole sheep, from the commencement of the operation.

LITERARY NOTICES. Lastly, we have 60 minutes : 213 minutes :: 1 sheep: 18, the

GREEK. part the wolf ate; 20 minutes : 11} minutes :: 1 sheep : 3, the part the tiger ate; 30 minutes : 14 minutes :: 1 sheep : 3, the The Third Volume of CASSELL'S CLASSICAL LIBRARY will contain the part the lion ate. Whence, it tal sheep, proof.

Acts of the Apostles in the original Greek, according to the text of Augustus

Hahn; with grammatical, historical, and expository Notes ; followed by a G. ARCHBOLD, St. Peter's. 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 [The following rythmical answer may please some readers.] with excellent materials for practice in translation. Mister Autodidactos, I've look'd o'er your rhyme,

LATIN. And propose now to tell you exactly the time

The first volume of CASSELL'S CLASSICAL LIBRARY is now ready, Which the Wolf, with the Tiger and Lioa combined,

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price Is. 68., containing Latin extracts for translation on the following Took to eat up the Sheep, and leave nothing behind.

subjects-Easy Fables, Mythology, Biography, The History of Rome, and Twenty-one and *wo-thirds are the MINUTES, you see,

Ancient Geography ; with a suitable Dictionary. The second volume,

which is publishing in weekly numbers price. 2d. each, will consist That the sheep was in eating among them all three.

of useful Latin Exercises, or English sentences, to be translated into Latin, Then as to the portion to each one allotted,

with numerous references to Andrews and Stoddart's Latin Grammar, This may, in the following words, be just noted :

valuable treatise now in the presen

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ON PHYSICS OR NATURAL PHILOSOPHY. . 18t 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 No. V.

described in the 2nd unit of time is 4 less 1, that is, 3; in the LAWS OF FALLING BODIES, INTENSITY OF

3rd unit it is 9 less 4, that is, 5; in the 4th unit, 16 less 9, that GRAVITY, &c.

is, 7; and so on. Hence, the spaces described in the 1st, 2nd, Falling Bodies. The three laws of Falling Bodies are the 3rd, 4th, &c., units of time are successively 1, 3, 5, 7, &c., following, which are only strictly accurate when the conside- according to the series of odd numbers. From this it is evident ration of the resistance of the air is omitted ; or, in other that the spaces described increase by equal quantities in equal words, when the bodies fall in a perfect vacuum.

times, which is in accordance with the definition already given Ist Law: All bodies, large or small, fall with equal rapidity

of uniformly accelerated motion. to the earth's surface, at the same place. This law is proved in a vacuum, and from heights in the atmosphere differing

The laws of falling bodies are only true when the bodies fall by the following experiment, called the guinca and feather expe. ) little from each other in comparison with the radius of the riment. Take a tube of glass of about two yards in length, and earth." When the bodies fall in the air, these laws are modi. a convenient diameter, fig. 11, closed at one of its extremities, fied by the resistance of the atmosphere; and when they fall and furnished at the other with a brass stop-cock; put into thi' tube, placed vertically, with the closed end lowest, any two from very unequal heights in the atmosphere, the force of bodies of different densities, such as lead and cork, gold and gravity is not strictly the same. paper, &c., and make a vacuum in it with

Galileo, an Italian philosopher and Florentine nobleman. Fig. 11.

an air-pump ; then quickly invert the was the first who made the discovery of these laws, and tube, by placing the closed end uppermost, where he was professor of the mathematics in 1611 A.D.

announced them to the students of the university of Pisa, and keep it in the vertical position; you will now see the light body and the heavy Inclined Plane. Various apparatus have been invented for body, such as the guinea and the feather, the purpose of proving the laws of falling bodies ; Galileo both fall to the other end of the tube employed the inclined plane in an original manner; Atwood with the same velocity. Readmit a little invented the machine known by his name; and M. Morin, air by opening the stop-cock, invert the director of the “ Conservatoire des Arts and Metiers" at Páris, tube in the same manner as before, and constructed an apparatus first proposed by M. Poncelet. you will see the light body falling more

An inclined plane is one which makes with a horizontal slowly than the heavy one, in proportion plane any angle less than a right angle. In proportion to the to its comparative weight. Lastly, re- smallness of the angle between these planes, so is the decrease aůmit the air completely, perform the of the velocity of a body which descends along the inclined same inversion, and you will find that plane. Thus, let AB, fig. 12, represent an inclined plane, A C the light body falls still more slowly than the horizontal plane, and pc a perpendicular to the horizontal before, in consequence of the greater effect

Fig. 13
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
difference in the action of gravity upon

?
different substances, for it acts alike upon
all substances, making them fall from the plane drawn from any point b in the inclined plane. If any
same height in the same time in a vacuum. body w rest upon this inclined plane, its weight P acting ver-
Moreover, under equal volume, all bodies tically may be resolved into two forces Q and ?, the one acting
experience the same resistance of the air in in a perpendicular and the other in a parallel direction to the
falling; but the force with which they are inclined plane a c. The first force, Q, will be completely
attracted overcomes this resistance in pro-counteracted by the resistance of the inclined plane which acts
portion to their mass.

in the direction QG, and the other force f only will act on the The resistance of the air to falling mass of the body n in order to make it descend along the plane. bodies is particularly evident in the case of liquids. When they fall in the air, line GP a length the number of whose units represents the

In order to ascertain the value of the force F, take on the they separate and fall in drops; but when weight P, and complete the parallelogram D GEH; then the they fall in a vacuum, they fall like a solid force f will be represented by the number of units of length mass, without separating into drops. This in GD. But the triangles D G H and A B C are similar, because phenomena is proved by the apparatus their angles are equal (Cassell's Euclid, Book VI., Prop. IV.); called the water-hariner; this is a tube whence we have of glass of about an inch in diameter, and about a foot or sixteen inches long, nearly half filled with water and hermetically sealed, after the air has been expelled that is, the force F Will be less than the weight P, in proportion by raising the water to the boiling point as the height B C of the inclined plane is less than its length A C.

When this tube is quickly inverted, the Thus we can make the force F as small as we please, by water in falling strikes against its lower end with a smart diminishing the height of the plane, or the angle" which it dry sound like that of the collision of two solid bodies.

makes with the horizon, and thus slacken the motion of the 2nd Law. The velocity acquired by a body falling in a moveable body M, so as to be able to take account of the spaces vacuum is proportional to the time of falling. Thus, at the described in one, two, three, &c., seconds, and this without end of 2, 3, 4, &c., times a given unit of time, the velocity altering the laws of the motion, since the force F is continued scquired will be 2, 3, 4, &c., times the velocity acquired in or constant. By such experiments as this, Galileo discovered that unit.

that the spaces described increased as the squares of the 3rd Law.

The spaces described by a body falling in a vacuum, are proportional to the squares of the times of falling. Atwood's Tachine.-The laws of falling bodies, however, Thus, if the times of falling be 1, 2, 3, 4, 5, &c., times a given were more clearly demonstrated experimentally by means of unit of time, the spaces described will be 1, 4, 9, 16, 25, a machine invented by Mr. Atwood, professor of chemistry in &c. times the space described in that unit.

the University of Cambridge. This machine is formed of a Since, according to the third law, the space described in the I narrow wooden piller, about seven: and a-half feet high, VOL. IY.

83

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GH: DG :: AB: BC, or

P: F :: AB : BC;

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m-f2m

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haring on the top & glass case, in which is placed a brass pulley (the weight m to fall by itself, now puts in motion this weight u, tig13. Over this pulleypasses a silken thread, so fine that its and the two other weights M and m. The quantity of motion, Height pred 1100 be taken into account, and having two equal or momentum, will therefore be still the same. If we denote

weights M and af sus the velocity of the mass at the end of a sęoond try x, the Fig. 13.

pended at its extremities. momentum will be (m+-2M); and by putting this equal to the
The axle of the pulley, momentem of m, when it falls alone, we have the equation
instead of resting on two
fixed bearings, is sup- (m-+211).xing; whence 2 =

Thus, if the weights
ported on the circumfe-
rences of four moveable M and a' were each 16, the weight m being unity or 1, we
wheels. By this arrange-
ment the axle of the should have 2: ; that is, the velocity of the mass would
pulley transmits its mo-
tion to the four wheels, be only one thirty-third part of the velocity which it would
and the sliding friction have if it fell freely in the air. By this means we can more
of fixed bearings is con- easily ascertain the nature of the force which causes bodies to
Ferted into the rolling fall, and also render the resistance of the air imperceptible.
friction of the wheels, a The first experiment performed by this machine proves that
contrivance by which the the spaces described by a falling body increase as the squares
friction of the axle is very of the times. The pendulum p being at rest, and the second
much diminished.

index being beyond zero, the weight M' is placed on the platB

On the pillar is fixed a form is and is loaded with the additional weight m, the whole clock-movement z, which being kept in the horizontal position by the extremity of the regulates a seconds' pen- lever D, and corresponding to zero on the scale. Removing

dulum p by means of an then the hollow stage B, and preserving only the final stage A, H

anchor escapement. This place the latter, by trials, at such a distance from the zero escapement is shown on point at I, that from this point to the stage a the weights » the dial-plate above the and take only one second in falling, the fall commencing at swing-wheel which occu- the instant when the pendulum having been put in motion pies the centre. This the index reaches zero on the dial-plate; for at this point the escapement oscillates with lever D is put in motion by the eccentric, and the platform I is the pendulum, and in- overturned, setting the weights m and i at liberty to fall. clining to the right and

Suppose now that we have found the height of the fall, or left alternately, at each the space descended in a second, to be seven divisions of the

oscillation, it allows one scale; then repeat the experiment as before, but remove the DR

tooth of the swing-wheel stage A to a distance from the zero point I, equal to four times to escane, The axis of the preceding distance, that is, to the 28th division of the this wlieel carries at its scale, and it will be seen that this space is described in anterior extremity an in, exactly two seconds by the two weights m and m'. In like dex marking seconds, and manner it will be found that at a distance nine times the first, at its posterior extremity, or at the 63rd division of the scale, the space will be described behind the dial-plate, an in three seconds; and so on.

The third law is, therefore, eccentric, shown at E on verified by experiment.

the left of the pillar. This In order to verify the second law by experiment, it must be P

eccentric moves with the recollected that in accelerated motion the velocity at a given index, and presses on a instant is that of the uniform motion which follows upon the lever D, which, by its accelerated motion. Hence, in order to discover according to inotion, overturns a small what law the velocity of a falling body varies, we have only platform I, employed to

to measure the velocity of the uniform motions which immesupport the mass mi'.

diately follow the accelerated motions successively after one, Parallel to the pillar, two, three, &c., seconds of the fall. and fastened to its base, The determination of the uniform motion after the acceleis a wooden scale of nearly rated motion is obtained by means of the stage B.

This is the same length, divided placed just at the distance from the zero of the scale which the into inches and tenths of two weights m and M' when descending reached in a second, as an inch, used for the in the first experiment; then, the additional weight m being purpose of measuring the

stopped in its descent by the stage B, the weight M' continues spaces described by the to descend alone, until it be stopped by the stage a, which is falling body. On this

placed below B at such a distance as that the weight'u shall scale are two stages A and B, which by means of tangent occupy only one second in passing from B to A. Now, from. I screws can be adjusted to any required height. The stage to B the motion is uniformly accelerated, and from B to A it is A is intended to receive the weight M' at the end of its course; uniform ; for the weight m being stopped by the stage B, and the stage B, which is hollow, allows this weight to pass gravity no longer acts from B to A, and the motion is only through it, and is used only to stop the progress of the addis cantinued in consequence of the inertia of the weight m'. The tional weight m which rests upon it at starting. The use of number of the divisions of the scale passed over by the weight Atwood's machine is to diminish the velocity of a falling M' from the one stage to the other will then represent the body, and to produce at pleasure a uniform motion, or a velocity acquired by the two weights m and m' at the end of motion uniformly accelerated.

one second. In order to understand the nature of this machine, suppose In repeating this experiment, the stage B is lowered to such that a small piece of brass m, which in the engraving rests on

a distance that the two weights m and i' take two seconds to the stage B, falls alone; let its velocity at the end of a second descend from the point i to the stage B; the stage A is then be denoted by g; its momentum or quantity of motion will lowered to a distance from B double of that at which it was in then be denoted by mg. If this piece of brass mn be placed on the first experiment. Thus, the two weights fall during two the veight x', when at the top of the scale, it will descend and seconds in a state of uniformly accelerated motion; then reachcommunicate part of its motion to the two weights M and '; ing the stage B, the weight M' alone passes over the interval for previously to this, the two weights being equal were in between the stage B and the stage 1. The velocity acquired equilibrium, the action of gravity in each being mutually at the end of two seconds is therefore double of that açquired balanced. It is plain that the same force which would cause at the end of one second. Similar experiments being made for

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three, four, &c., seconds, it will be found that the velocities | to guide a long wooden ruler which is applied to the cylinder acquired are three, four, &c., times the velocity acquired at the and is used to trace on its surface two kinds of equidistant end of the first second ; and thus the second law is verified,

lines, the one in planes perpendicular to the axis of the M. Morin's Apparatus. -- In this apparatus, or continned cylinder, and the other vertical. indicator of motion, the uniform rotatory motion of a cylinder The cast-iron piece, ar monkey, 1, guided in its descent by covered wi paper is combined with the motion of a falling two straight iron wires, F and G, firmly fixed at their extremibody, in such a manner that by means of a pencil properly ties, is placeri at first in a catch at 1), which can be opened at adjusted for the purpose, it describes on the paper a curve pleasure by drawing the wire z, to this monkey a is which represents the law of the motion. In fig. 14, the cylin. fastened to the penci} which describes, during its descent, der A, which is covered with paper, is about 9. fcet in height, the curre se on the cylinder as it revolves. From the forra and about 16 inches in diameter; this cylinder is set in motio:1 of this curve the 1:2 48 of motion are deduced. by a weight l, and this motion is communicaied by means of a For the space passed over by the pencil at the end of any cord to the drum B; this drum, by means of two hevelled given time, is at the point nof the curre equal to the portion wheels, communicates the motion to a rod H and to a wheel law of re vertical iraced on the surface of the cylinder. But and pinion i and o, which put the cylinder A in motion. ! the motion of the silird-r being ani!orn, we can take for the

The weight p having a tendency to accelerate its mo:ion duration of the full, when the moveable has descended to in, during its descent, M. Wagner, the maker of the apparatus, en- ! the are 1.1., bitis (en he point it and the vertical which is

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ployed, for the purpose of regulating the motion of the ruin is, rieithin ihrough the point at the origin, or tlie beginning of the å regulator of which the mechanism is concealecl in re 11:21 re. motion of the pencil

. In like manner, at any other point m; It is known in mechanics, however, by the name of the diff of the curve, the space passed over is represented by a man rential motion, and it depends both on the motion of a pendulum and the time by h'n. Now, by comparing the lengths am C, and of a fly furnished with leaves, which moves with great and a' in' with the arcs n m and I' 22', re find that the lengths rupidity. This fly is contained in a drum T, which rises or distances a w and a' n' are to one another as the squares of or falls according to the velocity of the apparatus. When their corresponding arcs ; thus it is clearly demonstrated that the motion is accelerated and the pendulum oscillates too the spaces passed over are to one another as the squares of rapidly, the drum rises, and the leaves of the fly then meeting the times of passing over ; and we therefore conclude that with the resistance of the air, the motion is retarded. On the the motion of falling bodies is one uniformly accelerated. other hand, when the velocity diminishes, the drum is lowered, The ratio which is found to subsist between the arcs hm, and the fly then meeting with less resistance from the air, the a'm', &c., and the verticals and mat, &fing shaw that the curve motion is accelerated. Thus & motion sensibly uniform is SR is a parabola whose axis is parallel to the generatrix of the for about 20 inches is sufficient.

plane the paper cover of the cylinder on which the curve is The wheel N, fixed on the axis of the cylinder, is employed l traced by the pencil

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