expressed by points, or by inverting the fraction which expresses Any set of numbers are said to be respectively proportional to the direct ratio. any other set containing the same number when the one set can A ratio is said to be compounded of two other ratios when it be obtained from the other by multiplying or dividing all the is equal to the product of the two ratios. Thus, is a ratio numbers of that set by the same number. Thus, 3, 4, 5 are compounded of the ratios and 1. proportional respectively to 9, 12, 15, or to , , . 3. Proportion. 7. To divide a given number into parts which shall be propor. Different pairs of numbers may have the same ratio. Thus, | tional to any given numbers. the ratios , 1, 3, are all equal. | Add the given numbers together, and then, dividing the given When two pairs of numbers have the same ratio, the four number into a number of parts equal to this sum, take as many numbers involved are said to form a proportion; and they them- of these parts as are equal to the given numbers respectively. selves, in reference to this relation subsisting among them, are EXAMPLE,- Divide 420 in proportion to the numbers 7, 5, called proportionals. Thus, 3, 4, 12, 16, are proportionals, and 3. because the ratio 1, or 3 : 4 = the ratio lâ, or 12 : 16. 7 + 5 + 3 = 15; A proportion is expressed either by writing the sign of equality And therefore the respective parts are-(=) between the two equal ratios, or by placing four dots in the is X 420 = 196. form of a square, thus, : : between them. * 420 = 140. Thus, the proportionality of 3, 4, 12, 16, might be expressed 420 = 84. in any one of the three following ways := 12; These parts are evidently in the proportion of 7, 5, and 3, 3: 4 = 12 : 16; 3:4 :: 12:16. and their sum, 196 + 140 + 84 = 420. The last expression would be read, 3 is to 4 as 12 is to 16. 8. The same method will apply if the given number or The first and fourth terms of a proportion are called the quantity is to be divided proportionally to given fractions. extremes; the middle two, the means. EXAMPLE.--Divide 266 into parts which shall be respectively 4. If jour numbers be proportional, the product of the extremes proportional to $, ;, and 1. is equal to the product of the means. Following exactly the same method as before, the answer, Take any proportion, 3:4::9:12, for instance. Expressing without reduction, would be this in the fractional form, we have = , and reducing these fractions to a common denominator 12 x 4, we get 3++ * 263, 7 + 8 + $ * 266, and 7 + + 7 * 266. 12 ~ 3 - 4 * 9 or 12 X 3 = 4 x 9. Or we may proceed thus :48 Reducing the fractions to their least common denominator, Now, 12 and 3 are the extremes, and 4 and 9 are the means, which is 60, we get 4:, !, and 18. of the given proportion. Conversely, if the product of two numbers is equal to the pro Now these fractions are proportional respectively to 40, 45, 48. dact of any other two numbers, the four numbers will form a Hence we have to divide 266 in the proportion of 40, 45, and proportion. Thus, since 48, to which the required answer is, since 40 + 45 + 48 = 133, 8 X 3 = 6 * 4 8, 4, 6, 3 form a proportion; 1 " X 266, 266, and 153 * 266, or, 8:4::6:3 or 80, 90, and 96. Or we may write it thus, 8:6:: 4:3; EXERCISE 41. Find in their simplest form :- 1. The ratio of 14 to 7, 36 to 9, 8 to 32, 54 to 6. Thus we see that either product may be separated to form 2. The ratio of 324 to 81, 802 to 99. the extremes, and that, the order of either the means or the 3. The inverse ratio of 4 to 12, and of 42 to 6. extremes being interchanged, the numbers still form a pro 4 Find the fourth term of the proportions, 3:5::6:-; 4:8::9:-; portion. }: :: :5. If three numbers be given, a fourth can always be found 5. Insert the third term in the following proportions—3:5:- :6; 4:8::-:9: 1: ::-: 2. which will form a proportion with them. 6. Insert the second term in the following proportions--3:-::5:6; This is the same thing as saying that if three terms of a proportion be given, the fourth can be found. 7. Insert the first term in the following proportions--:3:: 5:6; Take any three numbers--3, 4, 5, for instance. Then we have -:4::8:9; -:}:: 1:%. 1 8. Find a fourth proportional to 2.13, 579, and 3:14159, correct to 3:4:: 5: fourth term. 5 places of decimals. Therefore 9. Divide 100 in the ratio of 3 to 7. 10. Two numbers are in the ratio of 15 to 3t, and the smaller is 75 ; 3 * fourth term = 5 x 4 (since the products of the means and find the other. extremes are equal). 11. What two numbers are to each other as 5 to 6, the greater of Therefore, dividing both of these equalities by 3— them being 240 ? *** As tests by which the correctness of the processes of Fourth term = 4, the required number. addition, subtraction, multiplication, and division may be ascertained, were given in Lessons in Arithmetic, II. to V., it Here we have found the fourth term, but we could in the same way find a number which would form a proportion with has not been thought requisite to give answers to the Exercises already given in abstract Arithmetic. the three given numbers when standing in any of the terms. The answers will, howFor instance, for the second term we should have ever, be supplied to future examples in concrete Arithmetic. 3: second term :: 4:5, and therefore MECHANICS.-IX. 4 X second term = 5 * 3. THE STEELYARD. Hence, dividing both of these equalities by 4 ANOTHER weighing instrument is the steelyard, which (Fig. 54) is a lever of the first order, to the short arm of which is attached at b a hook from which the substance, w, to be weighed is and similarly for the other two terms. suspended, while on the long arm slides the movable counter. The most important application of proportion is the solution poise p. The object aimed at in this instrument being that a of examples of this kind, where three terms of a proportion are small weight, P, should balance a large one, w, on the hook, it given to find a fourth. This is what is usually called Rule of is clear that there must be a corresponding disproportion in the Three, which will be dealt with in a future lesson. arms—the fulcrum, a, must be near one of the ends of the beam. 6. It is evident that if the two terms of a ratio be multiplied Further, since it is necessary that the steelyard should take an or divided by the same quantity, the ratio is unaltered. | horizontal position, both when loaded and unloaded at its hook, second term w ry na Planan pa ring fan the form a nal * many of man weight, being hitiy tak tie the other, #puppertoning by the park na uby Martine gewaart vanntan, together , malert the ** indes. zm.** is heerily wanted it a kve will be bra mwamu, ne kun tret moitastant my ha distanca and, the centre of gravity of the whole lete has being zeszly at some point 5. st . On some sostanee, to be Aventna vin the wait, Arvintim ja w to w mnltiplier lrg ah, weighert, being suspended Ninn, in tay the great, Ind, nu mopone me pennd and w from x, the index muvas frus marinh nen w* how in namkuwa wewyn times a, b ennal ones ita zero point, Q. tp the quad. the tante in the nambarinnar frono f7, which talle on the exact mant until the weight of the fritten thrlin 7 pewna en the work, namely, that you find it by lever acting at & balances w H arring Trinh t, to the left key en rak ranh anal tab. lat w be 19 para ir % minde, then in like manner you ments of these forces are mananes 14 en pressu epal to ah. It thux appears that the equal, which will be when w Wwwviwme for the subwa ya panda are semnal to ench other, multiplied by B = is equal to and we thay Wardie lag Arwn the following role for gruinating the weight of the lever al Pig. 56. tiplied by GL The divisions in Me the mart kubivision by thringing the unloaded in of the quadrant corresponding Mament inter non burriumtul ilin Iry the counterpoine. Put to the several weights 1, 2, 3, 4, ete., suspended from Bars, thalt in the heak, or in the pian, much i number of even pounds however, best determined by experiment for each weight. ** will prah the suminterprue to the greatest distance it can go emlarm for event funda, and divide the distance between this THE LEVER WHEN THE FORCES ARE NOT PARALLEL Inut premature and the same point tato nn many equal parts an In all the cases of lovers and weighing instruments we have Whapo ure than judam the hook. The points of divinion so so far considered, the forces were supposed parallel-in weighing chained are the position of the counterpoine for the moveral instruments necessarily 86. The treatment of the subject is, pomnia mpe bar that number, however, not complete until the condition of equilibrium is de I'm bull and gaiter pounde thone divisione must be mub | termined for levers the forces acting on which are not parallel. divide, and from aronter weights than one pound will balanco | This is the most general case that can occur, and indeed it w the long urin, the cuminterpolne must be doubled or troblod, includes all the others. To Cearly understand it, let a lever be If the moolyard be intended for weighing small objects, defined a mass of matter of any shape which has one fixed paid much na lettera, the counterpolna may be ounoon, or tonths in it. It may be a bar straight, or simply bent, or bent and it antenne, or even finallor wolghta, an ooonnion requiros. twisted, or it may be a solid lock. So long as there is one 1 Move Alpare that the construotion of a stoolyard is very point fixed, wo may treat it as, lever, that point being the simple, and that any handy pornon of a mechanionl turn / fulorum. may make one of stool or iron, Moreover, the two forces whichaot on it are supposed to be or oven of piece of hard such that their directions when rodaced meet, and that their wood, without much troublo plano passes through the fulcru,. In cases where the two foroon do not moot, or their plan does not pass through the TIN DANIH NALANCE. fulorum, there cannot be equilibum. For example, the outThis is a species of stool | utrotohod right arm of a man is a Fer, of which the fulcrum is Parl, in which (Fig. 65) the in the right shoulder. Suppose, ane stretches it before him in Pulorum immovable, and the a horizontal position, one force is plied to the hand obliquely counterpoine in the weight of from him towards the left to th ground, while another acts the beam moting at its centre horizontally at his elbow towards e richt and at right angles of w rity, the autuatanve to be weighed being mumponded to the arm; those foroes cannot set and therefore would not # hotell our placed in man, at the extremity, A, on under any circumstanoes keep them in equilibrium; further, the hep all of the rulorum. The question is, how may you even were they to meet, they wont su keep it unless their ****dwie woh all humbumour To do this, let us suppose plane passed through the fuloru in the shoulder seket. SapPhan um fer www pounds and that I ounce of some substance posing the forces, therefore, the desired mately, that In Wurout in the evalo, then it is evident that the fulorum, their directions meet and their Lenasses through the fulcrum, E what is the condition of equilibrium ? In order that you may the perpendicular from o on A P. So likewise is the moment clearly understand this, the knowledge of the following geo- of A Q in reference to o equal to AQ multiplied into o y, the metrical principles is necessary. corresponding perpendicular. What I have then to prove is FURTHER PROPERTIES OF A PARALLELOGRAM AND TRIANGLE. that these products are equal. But they are equal; for, from 1. The area of a triangle is half that of any parallelogram which the second geometrical has its base for one side, and a line drawn through its vertex parallel principle above, the to that base for the side opposite.--This areas of the triangles appears from Fig. 57, where A V B is the A OP, A OQ, are half triangle, and A B C D any parallelogram these products; and, by the third, since on A B formed by drawing from A and B any two parallel lines AD, BC to meet the these triangles stand parallel D c to AB through v. For, draw Fig. 60. on the common base VE parallel to AD, and therefore parallel AO, and the line PQ to BC, to meet AB in E. Then the triangle jo La joining their verticos, being a diagonal, is bisected by A R, that is, AV B is made up of the two triangles A VE by that base, their areas are equal. The moments of a P and a Q, and B V E. But since A E V D is a parallelo therefore, in reference to o are equal, as I undertook to prove. gram, the triangle A V E (Lesson III.) is Now, to apply this to the lover, using the same figure, let us Fig. 57. equal to ADV, and is therefore half the suppose the two forces to be a P, AQ, meetings is therefore half the suppose the two forces to be A P, AQ, meeting, as I have stated parallelogram A EV D. So likewise is B VE to be necessary, at some point A. Then it is evident, since there half BEVC; and therefore the triangle A V B half ABCD. is but one point fixed in the body, that there cannot be equi. 2. The area of a triangle is, in numbers, half the product of its librium unless the resultant of AP and A Q passes through that base and the perpendicular from its vertex on that base.—This point, and is there resisted by the supports that fix it. The follows from the previous principle. Let the number of inches fulcrum, therefore, you see, must be on the resultant, and thereor feet, say inches, in A B (Fig. 58) be 6, and in the perpen fore taking o to be the fulcrum, we must have A P multiplied into dicular, v E, be 7, and construct on A B a parallelogram, ox equal to A o multiplied into o y, that is, ABCD, whose sides are parallel to this perpendicular. Such a the moments of the forces in reference to the fulorum must be equal. parallelogram is termed a "rectangle," on account of its angles We arrive being all right angles. Mark out the inches | thus at the two following modes of stating On A B and V E, and draw the dotted lines in the condition of equilibrium in a lever, at either of which may be selected for use as the figure parallel to A B and v E, cutting this rectangle into the smaller ones the sides the occasion requires :of which are all equal to one inch, and which 1. In a lever, the forces not being parallel, are therefore so many square inches. Now the power multiplied by the perpendicular there are seven rows of these squares, one row from the fulorum on its direction is equal above the other, and there are six squares in to the resistance multiplied by the perpen Fig. 61. each row; and therefore there are altogether dicular on its direction. 7 times 6, or 42, square inches in the rectangle. 2. The power and resistance are to each other inversely as the But the triangle being half the rectangle, isa perpendiculars dropped from the fulcrum on their respective half of 42 square inches, that is, it is, in num. Fig. 58. directions. THE WHEEL AND AXLE. bers, half the product of the base and perpendicular. Were the numbers 13 and 9, or any other pair whatever, . This useful mechanism, of which several forms are given in the reasoning would be the same. - Figs. 61, 62, and 63, is a kind of lever, or succession of levers, 3. If two triangles stand on opposite sides of a common base, revolving round an axis, from which they project at right angles. and the line joining their vertices is bisected by that base, the Corresponding to this central axle triangles have equal areas.-In Fig. 59, the triangles A B C, ABD line is a cylindrical axle of some stand on the common base, A B, at opposite sides, and D o join thickness, round which winds the ing their vertices is supposed to be bisected at m; I have to rope which bears the resistance, or prove that the areas of the triangles are equal. Draw E F and weight, to be raised. In Fig. 61 is the simplest form of the instrument, HG through A and B parallel consisting of an horizontal axle and toDc, and also through D and c draw 1 E and G F parallel four levers, which are worked in to AB. Then we have a succession by the power. In the ship’s capstan for raising the anchor large parallelogram E F G H, which is divided into four Fig. 62, (Fig. 62), the resistance acts horizonsmaller ones by A B and D c. tally, a man pushing also horizontally But since DC is bisected at the end of each lever, the power being multiplied in the proat m, making mc equal to portion of the number of levers and men. We have in Fig. 63 Fig. 59. MD, and therefore A E equal another form, where the levers are the spokes of a wheel, and the to A F, the parallelograms power A works in succession on them along APGB and AEHB are equal to each other. But, as proved the tire as they come round. above, the triangles ABC and A B D are half of these parallelo The principle in all is the same, whether grams, and therefore are also equal to each other, as was the resistance and power be parallel or not, required to be proved. and may be understood from Fig. 64, which We now return to our Mechanics, applying these geometrical represents a transverse section, the outer principles to determine circle being the wheel and the inner the axle. The central line of the axle, which THE MOMENTS IN THE LEVER OF FORCES NOT PARALLEL. you must conceive perpendicular to the Two such forces, A P, AQ (Fig. 60), being supposed to meet at paper at the centre of these circles, is the kome point, A, to which they are transferred, and there com fulcrum, represented by the point o. The pounded into a resultant A R, represented by the diagonal of the line A B thus is seen to be the lever, at the Tarallelogram, APBQ, and o being a point taken at random on ends of which the power, P, and resistance, ОА that diagonal, we can prove the following proposition : w, act; and, as already proved, these forces The moments of two intersecting forces in reference to any point Fig. 63. must be inversely as o A to O B, which lines on their resultont are equal to each other.--Now the moment of a are the radii of the wheel and axle respectively. When the power Torce in reference to a point, as has been already explained, is and resistance act parallel to each other this is evident; but the the product of the force by the perpendicular dropped on it from same holds good were they not so to act, as in the capstan, where that point. In Fig. 60, therefore, the moment of AP in refe- the power is continually changing direction as the sailors go round; se to 0, 2 point on the resultant, is a P multiplied into ox, for, referring again to Fig. 64, if the power were to act not in the W line A P, but along any other tangent to the large circle, the per LESSONS IN GERMAN.-XXI. pendicular from the fulcrum o on its direction would still be the radius of the wheel ; and, by the general principle of the lever SECTION XL.-PECULIAR IDIOMS—(continued). established in this lesson, the power and Was für ein ( 66.5), literally, what for a, answers to the English resistance would be still inversely as the "what kind of," or simply “what;" as :-Was für ein Buch baben radii of wheel and axle. Sie? what kind of a book have you? Was für ein Messer ist das? A treadmill, used for punishment in prisons, what kind of a knife is that? Für, in this connection, loses its is another instrument of this kind, the power prepositional character, and may precede any o being the weight of the prisoners ascending Bücher fint ties? what kind of books are these ? Was für Bücher the steps placed on the outside of the wheel, haben Sie? what kind of books have you ? Mit was für einem and the resistance the weight of the water Buche sind Sie beschäftigt? with what kind of (a) book are you em. pumped, the corn ground, or other work done. The windlass is another, turned generally by l i. Was für is likewise used in the way of exclamation, correFig. 64. a winch handle, and used to raise water from sponding to " what," as :-Was für Thorheit! what folly! Was wells, or lift goods into stores. In Fig. 21 für ein Mann! what a man! med abbrevini für ein Mann! what a man! Welch, abbreviated from welcher, is (page 188) the reader will find an example of the utility of the used in the same manner as dit ein Mann' what a man! wheel and axle as a mechanical power in the crane, by which two 2. Seter and jeglicher are often preceded by the indefinite article, men, by turning the winch-handle attached to the axle, are able to the axle, are able and are then, accordingly, inflected after the Mixed Declension. to lift a horse out of the steamer alongside of the quay. (Sect. X.) They are never used in the plural, as :- Der Tod jetes A particular form of the windlass, which was first invented in Menschen, or eines jeden Menschen ist gewis, the death of every man is China, and which may therefore be called the “Chinese windlass," certain. Gin Zeber mus sterben, every one must die. is given in Fig. 65, where only the axle is represented, consisting 3. Aller, unlike the English “all," is joined directly to its of two parts, one thicker than the other, but both forming one noun withont any article intervening. as: Auer Wein all the solid piece. The winch handle, wine. Alles Wasser, all the water, etc. or wheel, is to the right con Our word "all,” when connected with the names of countries, nected with the larger axle. The towns, etc., as also in such phrases as “all day, all the time, weight to be raised is suspended all my life," etc., is not expressed in German by all, but by from a hook attached to a pul. ganz, as :—Han; Guropa, all Europe. Banz Böhmen, all Bohemia. ley, round which the lifting rope Die ganze Schweiz, all Switzerland. Den ganzen Tag, all the day, or passes, one part winding round the whole day. Die ganze Zeit, mein ganzes Leben, etc. the thick axle while the other Alle or all, in some elliptical phrases, is equivalent to our "all unwinds from the thin. The gone," " no more," and the like, as :-Sein Geld ist alle, his money weight with each turn of the is all gone. wheel ascends by the difference 4. Mancer answers to “ many a," as :-Mancher Reiche ist uns between the length of the rope glüdlich, many a rich man is unhappy. that winds and unwinds, that 5. Solcher is often preceded by the indefinite article, as also is, by the difference between by fein, and is then, like jeter and jeglicher, inflected after the the circumferences of the Mixed Declension, as :-Gr ist eines solden Lebens nicht würdig, he is two axles. Moreover, since not worthy of such a (a such) life. So babe fein solches Buch, I the weight is equally divided have no such book. between the two ropes which Fig. 65. 6. Aller, mancher, solcher (and welcher, see R. 1) often drop the ascend from the pulley, the last syllable, and are then undeclined. Thus, aller, when it preforce acting at the circumference of each axle is half the weight. cedes a pronoun, is often abbreviated to all; mancher, when it It is evident, moreover, that the power applied to the winch precedes an adjective, often becomes manch; solcher (as also handle has to balance the difference of the actions of these welcher) is always thus abbreviated when it precedes the indefi. forces at the axle, or the moment of the power must be equal to nite article, as also, sometimes, when it precedes an adjective, the difference of the moments of these forces. But each force as :-3dh habe all mein Geld verloren, I have lost all my money. being half the weight, its moment is half the weight multiplied Ich habe all diese Bücher gekauft, I have bought all these books. by the radius of the axle at which it acts; and therefore their Mand cbrlıder Mann ist arm, many an honest mari is poor. difference is equal to half the weight multiplied by the difference Solch ein Tag ist angenehm, such a day is agreeable. Solo of the radii of the axles, or, which comes to the same thing, icônes Papier ist theuer, such beautiful paper is dear, etc. It to the weight into half the difference of these radii. But the should however be noted, that, as in the above examples, when moment of the power being that force into the radius of the the abbreviated form is followed by an adjective, this latter, inwheel, we immediately learn that stead of being inflected after the New Declension (Sect. IX. 2), In the Chinese windlass the power multiplied follows that of the Old ($ 29). by the radius of the wheel is equal to the resist- 7. Giniger and etlicher are regularly declined. They are ance multiplied by the difference of the radii nearly synonymous, and answer to our words “ some, a few, of the axles. etc., as :-Er sprach nur einige Worte, he spoke only a few words. THE COMPOUND WHEEL AND AXLE. Gr hat noch etliche Freunte in Teutschland, he has still some friends in Germany. Er wohnt in einiger Entfernung von der Stact, he reThis is a combination of wheels and axles, of sides at some, or a little distance from the city. Nach einiger Zeit the kind already explained, made for the same fam er, after some time he came. Ich habe noch etliches Mehl, I still purpose as the similar combination of levers in have got some flour. Gtliches fiel an den Weg, some fell by the war. Lesson VIII., namely, the mechanical advan- | side (Mark iv. 4). tage of a multiplication of the effect of the 8. Etwas, besides the signification noticed in Sect. XIV. 2, has power. The wheel and axle being once clearly also an adverbial use, and answers to "somewhat," as :-CT 1 understood to be a lever, there can be no etwas älter, als id, he is somewhat (or something) older than 1. difficulty in extending the rule which holds 6s ist etwas fälter, als vorgestern, it is somewhat colder than the day Fig. 63. good of the compound lever to this combina- | before yesterday. tion. In Fig. 66 is such a combination. Ry VOCABULARY. cogged teeth the axle of each wheel works on the circumference of the next succeeding, the power, p, being applied by a rope to Ab'legen, to lay aside. | Beschwer're, f. hard- | Blind, blind. the circumference of the first wheel, which does not require teeth. which does not require teeth. Anblid, m. aspect. T ship. Darů ber, about it It is evident that, as explained of the compound lever, the condi An'näherung, f. ap- Vesik'en, to possess. thereon. tion of equilibrium must be that proach. Bewun'derung, f. ad. Davon', of it, thereIn the compound wheel and axle, the power is to the resistance Vege'hen, to commit. miration. of. as the product of the radii of the axles is to the product of the Beschäftigen, to em- Bilten, to form, con. Dennoch, still not medii of the wheels. withstanding ploy. stitute. |