expressed by points, or by inverting the fraction which expresses the direct ratio. A ratio is said to be compounded of two other ratios when it is equal to the product of the two ratios. Thus, is a ratio compounded of the ratios and J. 3. Proportion. Different pairs of numbers may have the same ratio. Thus, the ratios,,, are all equal. Any set of numbers are said to be respectively proportional to any other set containing the same number when the one set can be obtained from the other by multiplying or dividing all the numbers of that set by the same number. Thus, 3, 4, 5 are proportional respectively to 9, 12, 15, or to }, 4, 1. 7. To divide a given number into parts which shall be proportional to any given numbers. Add the given numbers together, and then, dividing the given number into a number of parts equal to this sum, take as many of these parts as are equal to the given numbers respectively. EXAMPLE.-Divide 420 in proportion to the numbers 7, 5, and 3. 7+ 5+ 3 = 15; When two pairs of numbers have the same ratio, the four numbers involved are said to form a proportion; and they themselves, in reference to this relation subsisting among them, are called proportionals. Thus, 3, 4, 12, 16, are proportionals, because the ratio, or 3: 4 = the ratio 3, or 12: 16. 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 form of a square, thus, :: between them. Thus, the proportionality of 3, 4, 12, 16, might be expressed in any one of the three following ways:— 1 = 1; 3:4 12:16; 34 12 16. The last expression would be read, 3 is to 4 as 12 is to 16. 4. If four numbers be proportional, the product of the extremes is equal to the product of the means. = Take any proportion, 3: 4 :: 9: 12, for instance. Expressing this in the fractional form, we have , and reducing these fractions to a common denominator 12 x 4, we get Thus we see that either product may be separated to form the extremes, and that, the order of either the means or the extremes being interchanged, the numbers still form a proportion. 5. If three numbers be given, a fourth can always be found which will form a proportion with them. This is the same thing as saying that if three terms of a proportion be given, the fourth can be found. Take any three numbers-3, 4, 5, for instance. Then we have 3:45 fourth term. Therefore × 420 = 196. These parts are evidently in the proportion of 7, 5, and 3, and their sum, 196 + 140 + 84 = 420. 8. The same method will apply if the given number or quantity is to be divided proportionally to given fractions. EXAMPLE.--Divide 266 into parts which shall be respectively proportional to,, and . Following exactly the same method as before, the answer, without reduction, would be3 x 266, 星++醬 Or we may proceed thus: × 266, and × 266. 3 + 1 + 1 Reducing the fractions to their least common denominator, which is 60, we get 18, 15, and 18. Now these fractions are proportional respectively to 40, 45, 48. Hence we have to divide 266 in the proportion of 40, 45, and 48, to which the required answer is, since 40 +45 + 48 = 133, 1 × 266, 266, and × 266, or 80, 90, and 96. EXERCISE 41. Find in their simplest form : 1. The ratio of 14 to 7, 36 to 9, 8 to 32, 54 to 6. 3. The inverse ratio of 4 to 12, and of 42 to 6. 4 Find the fourth term of the proportions, 3:5::6; 4:8::9: -; 1:1: 6; 5. Insert the third term in the following proportions-3:5::4:8:9; } } :: ~ : }. 6. Insert the second term in the following proportions-3:-:: 5:6; 4:8:9; -8. 7. Insert the first term in the following proportions- -:35:6; -:48:9; -:::: 8. 8. Find a fourth proportional to 2.13, 579, and 3.14159, correct to 5 places of decimals. 9. Divide 100 in the ratio of 3 to 7. 10. Two numbers are in the ratio of 15 to 34, and the smaller is 75; 3 x fourth term = 5 x 4 (since the products of the means and find the other. extremes are equal). Therefore, dividing both of these equalities by 3 11. What two numbers are to each other as 5 to 6, the greater of them being 240 ? As tests by which the correctness of the processes of addition, subtraction, multiplication, and division may be ascertained, were given in Lessons in Arithmetic, II. to V., it already given in abstract Arithmetic. The answers will, how has not been thought requisite to give answers to the Exercises ever, be supplied to future examples in concrete Arithmetic. MECHANICS.-IX. THE STEELYARD. 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 ba hook from which the substance, w, to be weighed is suspended, while on the long arm slides the movable counterpoise P. The object aimed at in this instrument being that a small weight, P, should balance a large one, w, on the hook, it is clear that there must be a corresponding disproportion in the arms-the fulcrum, a, must be near one of the ends of the beam. Further, since it is necessary that the steelyard should take an horizontal position, both when loaded and unloaded at its hook, it is essential that its own centre of gravity should lie somewhere on the short arm; for then the counterpoise can balance it when placed in some position on the other arm, such as that marked 0, in the figure. For this reason steelyards are made heavy at one end. 13 12 11 10 9 8 6 54321 To Graduate a Steelyard.—The centre of gravity of the beam being on the hook side of the fulcrum, let it be brought into an horizontal position, no weight being on the hook. Then, as proved in Lessons VII. and VIII., the moment of P is equal to the moment of the beam, that is, the weight of the beam multiplied into the distance of its centre of gravity from a vertical line through the fulcrum, is equal to P multiplied into the distance of o from that line. At the point o so found draw a line across the beam; that line represents the zero division of the long arm, or the division at which P produces equilibrium, the weight on the hook being nothing, cipher, or zero. Р W Fig. 54. Now, supposing that any number of pounds, w, of any substance are hung on the hook, while P is shifted to the left until, as in the figure, the arm is again horizontal, we have P multiplied by the distance of its ring from the fulcrum a equal to w multiplied by ab (this line ab being supposed horizontal), together with the moment of the beam. But P multiplied by the distance of the zero division from a, is equal to the moment of the beam, as already proved; therefore it follows that P multiplied by its distance from the zero division is equal to w multiplied by a b. Now, in order to graduate, let us suppose P one pound and w seven. Then we have in numbers seven times a b equal once the distance of the counterpoise from o, which tells us the exact position of P for 7 pounds on the hook, namely, that you find it by measuring from o to the left seven pieces each equal to a b. Let w be 13 pounds or 3 pounds, then in like manner you measure 13 or 3 pieces equal to a b. It thus appears that the subdivisions for the successive pounds are equal to each other; and we may therefore lay down the following rule for graduating a steelyard: Find first the zero subdivision by bringing the unloaded instrument into an horizontal position by the counterpoise. Put then on the hook, or in the pan, such a number of even pounds as will push the counterpoise to the greatest distance it can go on its arm for even pounds, and divide the distance between this last position and the zero point into as many equal parts as there are then pounds on the hook. The points of division so obtained are the positions of the counterpoise for the several pounds up to that number. must be shifted to the point in which FR is to F P in the proportion of 16 to 1, there being 16 ounces in the pound. This comes to dividing the distance R P (which is known) into seventeen equal parts, as proved in Lesson IV., and taking the first point of division next to P for the fulcrum. If there be 2 ounces in the pan, RF must be to F P as 16 to 2; that is, you divide RP into 18 parts, and take the fulcrum 2 from P. If there be 7 ounces, you divide into 23 parts, and take 7 next to P; and so on for all the ounces from 1 to 16 you may determine the several positions of the fulcrum, marking them as you proceed. If the beam be of any other weight, you follow a similar course, dividing R P into as many equal parts as there are ounces in the sum of the weights of the beam and substance, and counting off as many divisions from P as there are ounces in the latter. From all this it is evident, first, that the subdivisions are not equal to each other, as in the steelyard; secondly, that the operation of graduation is more troublesome than in that instrument. The Danish balance, however, has the advantage of not being encumbered with a movable counterpoise; it carries its own imperial standard weight within itself. THE BENT LEVER BALANCE. The principle of this instrument, a species of which is largely sold for weighing letters, may be understood by the aid of the accompanying Fig. 56. On an upright stand is placed a quadrant arc, Mo, of which c is the centre. Round c as a fulcrum revolves a lever, usually bent, but in the figure represented as formed of two arms at right angles to each other. The arm CB is generally of small weight, being lightly constructed, while the other, CG, called the "index arm," is heavily weighted at its lower end, the centre of gravity of the whole lever thus being nearly at some point, G, on that arm. On some substance, w, to be weighed, being suspended from B, the index moves from its zero point, o, up the quadrant until the weight of the lever acting at G balances w at B, that is, until the moments of these forces are equal, which will be when w multiplied by B H is equal to the weight of the lever multiplied by G I. The divisions of the quadrant corresponding to the several weights 1, 2, 3, 4, etc., suspended from B are, however, best determined by experiment for each weight. Fig. 56. THE LEVER WHEN THE FORCES ARE NOT PARALLEL. In all the cases of levers and weighing instruments we have so far considered, the forces were supposed parallel-in weighing instruments necessarily so. The treatment of the subject is, however, not complete until the condition of equilibrium is deFor half and quarter pounds these divisions must be sub-termined for levers the forces acting on which are not parallel. divided; and for greater weights than one pound will balance on the long arm, the counterpoise must be doubled or trebled, ctc. If the steelyard be intended for weighing small objects, such as letters, the counterpoise may be ounces, or tenths of an ounce, or even smaller weights, as occasion requires. It thus appears that the construction of a steelyard is very simple, and that any handy person of a mechanical turn may make one of steel or iron, or even of a piece of hard wood, without much trouble. P Fig. 55. F R THE DANISH BALANCE. This is a species of steelyard, in which (Fig. 55) the fulcrum is movable, and the counterpoise is the weight of the beam acting at its centre of gravity, P, the substance to be weighed being suspended from a hook or placed in a pan, at the extremity, R, on the other side of the fulcrum. The question is, how may you graduate such an instrument ? To do this, let us suppose the beam to weigh 1 pound, and that 1 ounce of some substance is placed in the scale; then it is evident that the fulcrum, F, This is the most general case that can occur, and indeed it includes all the others. To clearly understand it, let a lever be defined a mass of matter of any shape which has one fixed point in it. It may be a bar straight, or simply bent, or bent and twisted, or it may be a solid block. So long as there is one point fixed, we may treat it as a lever, that point being the fulcrum. Moreover, the two forces which act on it are supposed to be such that their directions when produced meet, and that their plane passes through the fulcrum. In cases where the two forces do not meet, or their plane does not pass through the fulcrum, there cannot be equilibrium. For example, the outstretched right arm of a man is a lever, of which the fulcrum is in the right shoulder. Suppose, as he stretches it before him in a horizontal position, one force is applied to the hand obliquely from him towards the left to the ground, while another acts horizontally at his elbow towards the right and at right angles to the arm; these forces cannot meet, and therefore would not under any circumstances keep the arm in equilibrium; further, even were they to meet, they would not so keep it unless their plane passed through the fulcrum in the shoulder socket. Supposing the forces, therefore, to be as described, namely, that their directions meet and their plane passes through the fulcrum, what is the condition of equilibrium? In order that you may clearly understand this, the knowledge of the following geometrical principles is necessary. FURTHER PROPERTIES OF A PARALLELOGRAM AND TRIANGLE. 1. The area of a triangle is half that of any parallelogram which has its base for one side, and a line drawn through its vertex parallel to that base for the side opposite. This c appears from Fig. 57, where A V B is the triangle, and A B C D any parallelogram on A B formed by drawing from A and B any two parallel lines A D, BC to meet the parallel D C to A B through v. For, draw E Fig. 57. B and B V E. VE parallel to AD, and therefore parallel to BC, to meet A B in E. Then the triangle A V B is made up of the two triangles AVE But since A E V D is a parallelogram, the triangle A V E (Lesson III.) is equal to A D V, and is therefore half the parallelogram AEV D. So likewise is B V E half BEV C; and therefore the triangle A V B half A B C D. 2. The area of a triangle is, in numbers, half the product of its base and the perpendicular from its vertex on that base. This follows from the previous principle. Let the number of inches or feet, say inches, in A B (Fig. 58) be 6, and in the perpendicular, v E, be 7, and construct on A в a parallelogram, ABCD, whose sides are parallel to this perpendicular. Such a parallelogram is termed a "rectangle," on account of its angles being all right angles. Mark out the inches on A B and v E, and draw the dotted lines in o the figure parallel to A B and v E, cutting this rectangle into the smaller ones the sides of which are all equal to one inch, and which are therefore so many square inches. Now there are seven rows of these squares, one row above the other, and there are six squares in each row; and therefore there are altogether 7 times 6, or 42, square inches in the rectangle. But the triangle being half the rectangle, is half of 42 square inches, that is, it is, in numbers, half the product of the base and perpendicular. Were the numbers 13 and 9, or any other pair whatever, the reasoning would be the same. A E Fig. 58. 3. If two triangles stand on opposite sides of a common base, and the line joining their vertices is bisected by that base, the triangles have equal areas.-In Fig. 59, the triangles A B C, ABD stand on the common base, A в, at opposite sides, and DC joining their vertices is supposed to be bisected at M; I have to prove that the areas of the triangles are equal. Draw EF and Н B HG through A and B parallel to DC, and also through D and c draw H E and G F parallel to A B. Then we have a large parallelogram E F G H, which is divided into four smaller ones by A B and D C. But since DC is bisected at M, making мc equal to M D, and therefore A E equal to AF, the parallelograms AFGB and A EHB are equal to each other. But, as proved above, the triangles A B C and A B D are half of these parallelograms, and therefore are also equal to each other, as was required to be proved. A Fig. 59. We now return to our Mechanics, applying these geometrical principles to determine THE MOMENTS IN THE LEVER OF FORCES NOT PARALLEL. Two such forces, A P, A Q (Fig. 60), being supposed to meet at some point, A, to which they are transferred, and there compounded into a resultant A R, represented by the diagonal of the parallelogram, A P R Q, and o being a point taken at random on that diagonal, we can prove the following proposition :The moments of two intersecting forces in reference to any point on their resultant are equal to each other.-Now the moment of a force in reference to a point, as has been already explained, is the product of the force by the perpendicular dropped on it from that point. In Fig. 60, therefore, the moment of AP in reference to o, a point on the resultant, is a P multiplied into ox, the perpendicular from o on a P. So likewise is the moment of A Q in reference to o equal to AQ multiplied into o Y, the corresponding perpendicular. What I have then to prove is that these products are equal. But they are equal; for, from X Fig. 60. the second geometrical principle above, the areas of the triangles AOP, AOQ, are half these products; and, by the third, since these triangles stand on the common base A O, and the line PQ joining their vertices, being a diagonal, is bisected by A R, that is, by that base, their areas are equal. The moments of a P and AQ, therefore, in reference to o are equal, as I undertook to prove. Now, to apply this to the lever, using the same figure, let us suppose the two forces to be A P, AQ, meeting, as I have stated to be necessary, at some point A. Then it is evident, since there is but one point fixed in the body, that there cannot be equilibrium unless the resultant of AP and AQ passes through that point, and is there resisted by the supports that fix it. The fore taking o to be the fulcrum, we must have A P multiplied into fulcrum, therefore, you see, must be on the resultant, and therethe moments of the forces in reference to o x equal to a Q multiplied into o Y, that is, thus at the two following modes of stating the fulcrum must be equal. We arrive the condition of equilibrium in a lever, either of which may be selected for use as the occasion requires : 1. In a lever, the forces not being parallel, the power multiplied by the perpendicular from the fulcrum on its direction is equal to the resistance multiplied by the perpendicular on its direction. Fig. 61. 2. The power and resistance are to each other inversely as the perpendiculars dropped from the fulcrum on their respective directions. THE WHEEL AND AXLE. Figs. 61, 62, and 63, is a kind of lever, or succession of levers, This useful mechanism, of which several forms are given in revolving round an axis, from which they project at right angles. Corresponding to this central axle line is a cylindrical axle of some thickness, round which winds the rope which bears the resistance, or weight, to be raised. In Fig. 61 is the simplest form of the instrument, consisting of an horizontal axle and four levers, which are worked in succession by the power. In the ship's capstan for raising the anchor (Fig. 62), the resistance acts horizontally, a man pushing also horizontally at the end of each lever, the power being multiplied in the proWe have in Fig. 63 portion of the number of levers and men. another form, where the levers are the spokes of a wheel, and the power A works in succession on them along the tire as they come round. Fig. 62, the resistance and power be parallel or not, The principle in all is the same, whether and may be understood from Fig. 64, which represents a transverse section, the outer circle being the wheel and the inner the axle. The central line of the axle, which you must conceive perpendicular to the paper at the centre of these circles, is the fulcrum, represented by the point o. The line A B thus is seen to be the lever, at the ends of which the power, P, and resistance, w, act; and, as already proved, these forces must he inversely as o A to O B, which lines are the radii of the wheel and axle respectively. When the power and resistance act parallel to each other this is evident; but the same holds good were they not so to act, as in the capstan, where the power is continually changing direction as the sailors go round; for, referring again to Fig. 64, if the power were to act not in the Fig. 63. line A P, but along any other tangent to the large circle, the perpendicular from the fulcrum o on its direction would still be the radius of the wheel; and, by the general principle of the lever established in this lesson, the power and resistance would be still inversely as the radii of wheel and axle. B A treadmill, used for punishment in prisons, is another instrument of this kind, the power being the weight of the prisoners ascending the steps placed on the outside of the wheel, and the resistance the weight of the water pumped, the corn ground, or other work done. The windlass is another, turned generally by a winch handle, and used to raise water from Fig. 64. wells, or lift goods into stores. In Fig. 21 (page 188) the reader will find an example of the utility of the wheel and axle as a mechanical power in the crane, by which two men, by turning the winch-handle attached to the axle, are able to lift a horse out of the steamer alongside of the quay. P A particular form of the windlass, which was first invented in China, and which may therefore be called the "Chinese windlass," is given in Fig. 65, where only the axle is represented, consisting of two parts, one thicker than the other, but both forming one solid piece. The winch handle, or wheel, is to the right connected with the larger axle. The weight to be raised is suspended from a hook attached to a pulley, round which the lifting rope passes, one part winding round the thick axle while the other unwinds from the thin. The weight with each turn of the wheel ascends by the difference between the length of the rope that winds and unwinds, that is, by the difference between the circumferences of the two axles. Moreover, since the weight is equally divided between the two ropes which ascend from the pulley, the Fig. 65. force acting at the circumference of each axle is half the weight. It is evident, moreover, that the power applied to the winch handle has to balance the difference of the actions of these forces at the axle, or the moment of the power must be equal to the difference of the moments of these forces. But each force being half the weight, its moment is half the weight multiplied by the radius of the axle at which it acts; and therefore their difference is equal to half the weight multiplied by the difference of the radii of the axles, or, which comes to the same thing, to the weight into half the difference of these radii. But the moment of the power being that force into the radius of the wheel, we immediately learn that In the Chinese windlass the power multiplied by the radius of the wheel is equal to the resistance multiplied by the difference of the radii of the axles. THE COMPOUND WHEEL AND AXLE. This is a combination of wheels and axles, of the kind already explained, made for the same purpose as the similar combination of levers in Lesson VIII., namely, the mechanical advantage of a multiplication of the effect of the power. The wheel and axle being once clearly understood to be a lever, there can be no difficulty in extending the rule which holds Fig. 66. good of the compound lever to this combination. In Fig. 66 is such a combination. By cogged teeth the axle of each wheel works on the circumference of the next succeeding, the power, P, being applied by a rope to the circumference of the first wheel, which does not require teeth. It is evident that, as explained of the compound lever, the condition of equilibrium must be that— In the compound wheel and axle, the power is to the resistance as the product of the radii of the axles is to the product of the radii of the wheels. LESSONS IN GERMAN.-XXI. SECTION XL. PECULIAR IDIOMS-(continued). Was für ein (§ 66. 5), literally, what for a, answers to the English "what kind of," or simply "what;" as :-Was für ein Buch haben Sie? what kind of a book have you? Was für ein Messer ist das? what kind of a knife is that? Für, in this connection, loses its prepositional character, and may precede any case, as::-Was für Bücher sind ties? what kind of books are these? Was für Bücher haben Sie? what kind of books have you? Mit was für einem Buche sind Sie beschäftigt? with what kind of (a) book are you em ployed? 1. Was für is likewise used in the way of exclamation, corresponding to "what," as:-Was für Thorheit! what folly! Was für ein Mann! what a man! Welch, abbreviated from welcher, is used in the same manner, as:-Welch ein Mann! what a man! 2. Jeter and jeglicher are often preceded by the indefinite article, and are then, accordingly, inflected after the Mixed Declension. (Sect. X.) They are never used in the plural, as:-Der Tod jeres Menschen, or eines jeden Menschen ist gewiß, the death of every man is certain. Ein Jeder muß sterben, every one must die. 3. Aller, unlike the English "all," is joined directly to its noun without any article intervening, as:-Aller Wein, all the wine. Alles Wasser, all the water, etc. Our word "all," when connected with the names of countries, towns, etc., as also in such phrases as "all day, all the time, all my life," etc., is not expressed in German by all, but by ganz, as-Ganz Europa, all Europe. Ganz Böhmen, all Bohemia. Die ganze Schweiz, all Switzerland. Den ganzen Tag, all the day, or the whole day. Die ganze Zeit, mein ganzes Leben, etc. all Alle or all, in some elliptical phrases, is equivalent to our gone," "no more," and the like, as:-Sein Geld ist alle, his money is all gone. 4. Mancher answers to "many a," as:-Mancher Reiche ist un glücklich, many a rich man is unhappy. 5. Solcher is often preceded by the indefinite article, as also by fein, and is then, like jeter and jeglicher, inflected after the Mixed Declension, as :---Er ist eines solchen Lebens nicht würdig, he is not worthy of such a (a such) life. Ich habe kein solches Buch, I have no such book. 6. Aller, mancher, solcher (and welcher, see R. 1) often drop the last syllable, and are then undeclined. Thus, aller, when it precedes a pronoun, is often abbreviated to all; mancher, when it precedes an adjective, often becomes manch; solcher (as also welcher) is always thus abbreviated when it precedes the indefi nite article, as also, sometimes, when it precedes an adjective, as:- -Ich habe all mein Geld verloren, I have lost all my money. Ich habe all diese Bücher gekauft, I have bought all these books. manch ehrlicher Mann ist arm, many an honest man is poor. Solch ein Tag ist angenehm, such a day is agreeable. Sel schönes Papier ift theuer, such beautiful paper is dear, etc. It should however be noted, that, as in the above examples, when the abbreviated form is followed by an adjective, this latter, instead of being inflected after the New Declension (Sect. IX. 2), follows that of the Old (§ 29). 7. Einiger and etlicher are regularly declined. They are nearly synonymous, and answer to our words "some, a few," etc., as-Er sprach nur einige Worte, he spoke only a few words. Er hat noch etliche Freunde in Deutschland, he has still some friends in Germany. Er wohnt in einiger Entfernung von der Stadt, he resides at some, or a little distance from the city. Nach einiger Zeit fam er, after some time he came. Ich habe noch etliches Mehl, I still have got some flour. Etliches fiel an den Weg, some fell by the way. side (Mark iv. 4). 8. Etwas, besides the signification noticed in Sect. XIV. 2, has also an adverbial use, and answers to "somewhat," as:-Er ist etwas älter, als ich, he is somewhat (or something) older than I. Gs ist etwas kalter, als vorgestern, it is somewhat colder than the day before yesterday. Seefisch, m. sea-fish. Gewähren, to grant, tude. Knochen, m. bone. able. to 24. Unter den Einwohnern sind manche sehr wohlhabend. 25. Haben Sie nicht auch schon manches Seltsame erlebt? 26. O ja, ich habe schon manches Merkwürdige erfahren. 27. Manch tapferer Soldat mußte in ter Schlacht sein Leben lassen. 28. Hat dieser Schriftsteller nicht viele gute Bücher geschrieben? 29. Gewiß, manche davon sind vortrefflich. 30. Haben sich die beiden Freunde über diese Sache verständigt? 31. Ja, in einigen Punkten sind sie miteinander übereingekommen. 32. Einige eng lische Schiffe gingen bei diesem Sturme unter. 33. Etliche Fluge Männer zogen sich aus ter Versammlung zurück. 34. Alle Einwohner der Stadt agree, accord. flüchteten sich bei der Annäherung der Feinde. 35. Manche Menschen brin Versammlung, f. meet-gen ihr ganzes Leben mit Nichtsthun zu. 36. War das Ihr Bruder, der ing. gestern den ganzen Tag in Ihrer Gesellschaft war? 37. Nein, es war mein Verständigen, to agree, Neffe, der mich alle Jahre einmal besucht. 38. Welch eine Größe hat vie to come to an ex- Erde, und wie viel kleiner ist ste dennoch, als die Sonne! 39. Welche Vor planation. züge hat der Mensch vor den Thieren? 40. Was für eines Vogels Feder Verwenden, to em- ift dics? 41. Ist der Schüler fleißig, so lernt er etwas. ploy, apply. Vortrefflich, excellent. Vorzug, m. advantage. Was für, what kind of. Werk, n. work. Wohl'habent, opulent. Zubringen, to spend, pass. EXERCISE 77. 1. Many a learned man has been misunderstood. 2. Oh, what folly does man commit in his life! 3. With what kind of society have you associated? 4. Many an industrious merchant has been ruined by an imprudent speculation. 5. Full many a flower is born to blush unseen [blühet im Verborgenen]. 6. Every leaf, every twig, and every drop of water, testifies infinite wisdom and power. 7. Every one must give an account of himself. 8. The whole environs of Coblentz are romantic. 9. All are well [wohl at home. 10. The conversation with such persons is instructive. 11. I have never heard of such an accident. What kind of a companion have 12. It is beautiful weather to-day, but somewhat colder than you? tion. Zurück ziehen, to retire, Such a storm I have not yet Such an emperor could thus yesterday. 13. I have had already many a pleasure. 14. I wish to have some lemons. 15. He came somewhat too late. IN these papers we shall not enter on the consideration of cryptogamic plants until we have noted the peculiarities that distinguish the different natural orders of flowering plants. Those which possess flowers are far more likely to arouse the young botanist's attention; they are more useful, and are those members of the vegetable world which botanists know most about. disap-Ranunculacea, as the one first to be considered. Let us see, We shall select the Crow-Foot tribe, termed by botanists then, in how few words a botanist defines the characters of Ranunculaceae : He spoke so softly, that I could Many a dream of youth pears with the years. After some minutes he re- The elephant is somewhat He was sick all the year. Er war das ganze Jahr krank. EXERCISE 76. 1. Was für Wetter ist es heute? 2. Es ist heute schönes Wetter, aber etwas kälter, als gestern. 3. Was für eine Meinung hegt er von dieser Sache? 4. Seine Meinung darüber ist nicht die beste (Sect. XXXV. 3). 5 Meine Gesellschaft ist ihm die angenehmste von der Welt. 6. Was für Biche sind dies? (Sect. XXXV. 3.) 7. Es sind Seefische. 8. Mit was für Arbeiten beschäftigt er sich? 9. Er beschäftigt sich theils mit Schreiben, theils mit Lefen. 10. Welch eine Macht hat die Musik über das Gemüth des lenschen! 11. Welch ein hoher Genuß ist es, die Welt zu sehen: 12. Welch einen herrlichen Anblick gewährt das Firmament mit seinen un zähligen Sternen! 13. Jeter Stern am Himmel biltet eine eigene Welt. 14. Der wahre Tugendhafte verwendet jeten Tag seines Lebens darauf, seine Fehler immer mehr abzulegen. 15. Hat nicht jeter Ihrer Freunte einen folchen Sut? 16. Nein, ein Jerer hat einen andern. 17. Solche Männer fine nothwendig, um das terland zu retten. 18. Haben Sie jenen Blinten gesehen, der eine Feinheit des Gefühles besitzt, die erstaunens martig ist? 19. Ja, ich habe ihn gesehen. 20. Der Geber eines solchen Geschentes ist zu loben. 21. Die Beschwerden einer solchen Reise stablen ten Körper. 22. Solche Handlungen werden tie Bewunterung der Nach weit hervorrufen. 23. So angenehme Stunden habe ich lange nicht gehabt. RANUNCULACEÆ. Characteristics.-Calyx polysepalous; petals hypogynous, in form various, sometimes absent; stamens ordinarily numerous; anthers usually adnate; carpels one or numerous, never combined; ovule reflexed; embryo dicotyledonous, small, at the base of a horny albumen; fruit apocarpous. A very pretty collection of hard names, is it not? and sufficiently unintelligible. Nevertheless, the reader, we are sure, will admit that if the characters of the Ranunculus, or CrowFoot tribe, admit of description in so few words, it is worth while to learn the meaning of these words. Well, then, let us set about it; let us analyse the definition clause by clause. First then: calyx polysepalous; what is the meaning of that? The reader, by this time, knows the meaning of calyx; it is the outside greenish-yellow whorl of which the buttercup flower is composed, and being made up of several parts (sepals, and the Greek word, woλus [pol-use], signifying many), the calyx is denominated polysepalous, a somewhat important characteristic thus easily conveyed in one word. Now for the second clause, petals hypogynous. As for the word petal, the reader knows its meaning already; but hypogynous, what is the meaning of that term? Complex words, like complex plants and complex animals, require dissection. Hypogynous being dissected into hypo and gynous, we shall soon arrive at its meaning. In the first place, hypo is an Anglicised form of the Greek word ɔ (hu'-po), under; and gynous is evidently a derivation from another Greek word yurn (gu'-ne), signifying woman. When, therefore, it is said that the petals are hypogynous, the sense meant to be conveyed is, that they spring from underneath the carpels or female parts of the flower. A very slight examination of a dis. sected buttercup will show that the arrangement of petals is as |