has informed me of his design; Pythagoras called those who love CORRESPONDENCE. ARITHMETIC. MR. ELITOR, Yesterday, for the first time, the POPULAR EDUCATOR was ut into my hands. At page 224, I find "Garçon Embarrasse."vishing for a solution of a certain question; but whether to clear up his own mind, or to embarrass the minds of others, we are not strictly 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. 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 multipli ation, you must place the multiplier under 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 rules is violated. 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 times the first, for 27; hence, the total product is that of 45,000. former, for 45,000; and the third partial product is that of three 900427-45927. In the third method, the first partial product is times the former, for 96; the third is that of 7 times the second for that of 12 times the multiplier, for 1,200; the second is that of 8 672,0000; 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, with out 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 ?" "What rule are you now in ? 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 whole numbers, and to add vulgar fractions, are different operations; and thus 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. Mr. 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 country-secrets otherwise unattainable. They are left to think that to add men, 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 servile 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 none; in a very barren land. But by some kind instinct I knew the principles of the fundamental rules in arithmetic before I knew their names. As years came on, and I began to see the work of others, I became a critic. A few of my pertinent remarks on teachers and writers on arithmetic I wili lay before you. I quarrelled first with the word MUST either expressed or implied. Huel Vor, near Heiston, Cornwall, "In subtraction you must when you borrow 1, carry ten to the next Sep. 19, 1853. THOMAS GUNDRY." SLOANE'S BALANCE. 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 penknife blade, easily procurable at any cutler's for a few pence, fixed 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., 114, Great Britain-street, Dublin. Sep. 17, 1853. SOLUTIONS. JOHN J. SLOANE, Thirteen thirty-sixths of the whole, the Wolf finish'd, 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 quille 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 themselves 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-Improve 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 instanta neous 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 conse quences. Hence, the man who has received the highest moral and religious education has the most tender conscience. When the conscience is sea. ed 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 the P. E. may 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 Bin 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. & EDWARDS (Swinton-st.): Thanks-R. 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 m re 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. "plution 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." 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 sheepsheep: 1 hour: 1 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: 1, the part the wolf ate; 20 minutes: 11 minutes :: 1 sheep, the part the tiger ate; 30 minutes: 13 minutes: 1 sheep, the part the lion ate. Whence, +H+4=38=1 sheep, proof. G. ARCHEOLD, St. Peter's. 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! there are sad flaws in Legendre's Geometry; don't forsake old Euclid; he has stood 2,000 years! The second case JENOBIA (Brighton): The "Historical Educator" is a substitute for the lessons in History in the P. E.-J. THOMPSON (Leicester): We are not certain. ingenious; many thanks for his kind endeavours on our behalf.-E. HART: J. F. ENTWISTLE (Wigan): His tables for the Octary Scale are very 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 SUBSCRI BER (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 minis try; mere spouting won't do. LITERARY NOTICES. 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 grammatical, 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 [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. The first volume of CASSELL'S CLASSICAL LIBRARY is now ready, price 1s. 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, & valuable treatise now in the press. ON PHYSICS OR NATURAL PHILOSOPHY. No. V. LAWS OF FALLING BODIES, INTENSITY OF Falling Bodies. The three laws of Falling Bodies are the Fig. 11. 1st Law: All bodies, large or small, fall with equal rapidity The resistance of the air to falling 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. Since, according to the third law, the space described in the VOL. IV. 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 1, 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. in a vacuum, and from heights in the atmosphere differing The laws of falling bodies are only true when the bodies fall 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 where he was professor of the mathematics in 1611 A.D. announced them to the students of the university of Pisa, 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 Paris, 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 вc a perpendicular to the horizontal plane drawn from any point n in the inclined plane. If any body м rest upon this inclined plane, its weight P acting vertically may be resolved into two forces a and F, the one acting in a perpendicular and the other in a parallel direction to the inclined plane a c. The first force, a, will be completely counteracted by the resistance of the inclined plane which acts in the direction a G, and the other force F only will act on the mass of the body м 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 DGEH; then the force F will be represented by the number of units of length in GD. But the triangles D G H and ABC are similar, because their angles are equal (Cassell's Euclid, Book VI., Prop. IV.); whence we have GH DG :: AB: BC, or that is, the force F will be less than the weight P, in proportion Atwood's Machine.-The laws of fall Fig. 13. THE POPULAR EDUCATOR. (m+2)=mg; whence x = mg m+2x having on the top a glass case, in which is placed a brass pulley | the weight m to fall by itself, now puts in motion this weight R, fig. 13. Over this pulleypasses a silken thread, so fine that its and the two other weights м and M'. The quantity of motion, weight need not be taken into account, and having two equal or momentum, will therefore be still the same. If we denote weights M and M' sus- the velocity of the mass at the end of a second by z, the pended at its extremities. momentum will be (m+2); 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 supported on the circumfeThus, if the weights rences of four moveabie M and M' were each 16, the weight m being unity or 1, we wheels. By this arrangement the axle of the should have x = pulley transmits its motion to the four wheels, and the sliding friction of fixed bearings is converted into the rolling friction 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 seconds' 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 the 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 M'. Parallel to the pillar, and fastened to its base, is a wooden scale of nearly into inches and tenths of ༡ 33 ; that is, the velocity of the mass would 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 spaces described by a falling body increase as the squares The first experiment performed by this machine proves that of the times. The pendulum P being at rest, and the second form 1, and is loaded with the additional weight m, the whole index being beyond zero, the weight M' is placed on the platbeing kept in the horizontal position by the extremity of the lever D, and corresponding to zero on the scale. Removing then the hollow stage n, 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 # and м 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 m and м at liberty to fall. the space descended in a second, to be seven divisions of the recollected that in accelerated motion the velocity at a given rated motion is obtained by means of the stage B. This i The determination of the uniform motion after the accele the same length, divided placed just at the distance from the zero of the scale which the an inch, used for the in the first experiment; then, the additional weight being two weights m and M' when descending reached in a second, as 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 ▲, which is scale are two stages A and B, which by means of tangent occupy only one second in passing from B to A. Now, from t falling body. On this screws can be adjusted to any required height. The stage to the motion is uniformly accelerated, and from B to A it is placed below в at such a distance as that the weight a' shall A is intended to receive the weight M' at the end of its course; uniform; for the weight m being stopped by the stage 3 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 addi- continued 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 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 a' at the end of 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 mg. If this piece of brass m be placed on the weight M', when at the top of the scale, it will descend and communicate 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 alanced. It is plain that the same force which would cause one second. descend from the point I to the stage B; the stage A is then 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 Apparatus. In this apparatus, or continued indicator of mot.cn, 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 r, and this motion is communicated by means of a cord to the drum B; this drum, by means of two bevelled wheels, communicates the motion to a rod H and to a wheel and pinion I and o, which put the cylinder a in motion. 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 piece, 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 м is fastened at R the pencil which describes, during its descent, the curve 8 R on the cylinder as it revolves. From the form 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 m of the curve equal to the portion am of the vertical traced on the surface of the cylinder. But the motion of the cylinder being uniform, we can take for the duration of the fall, when the moveable has descended to m, the arc hm, between the point m and the vertical which is Fig. 14. The weight P having a tendency to accelerate its motion during its descent, M. Wagner, the maker of the apparatus, em ployed, for the purpose of regulating the motion of the drum B, a regulator of which the mechanism is concealed in the figure. It is known in mechanics, however, by the name of the differential motion, and it depends both on the motion of a pendulum c, 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 drawn 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 m" and the time by h'm'. Now, by comparing the lengths a m and a'm' with the arcs hm and 'm', we find that the lengths or distances a m and a'm' 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 hm, 'm', &c., and the verticals am, a'm', &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 |