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France? 12. I long to be there. 13. Does not your mother

tarry too long ? 14. She is very long in coming. 15. Have SECTION LVII.-IDIOMATIC PHRASES.

you changed the forty-franc piece ? 16. I have not changed it 1. CHANGER [1, see § 49 (1)], used in the sense of to change, to yet. 17. Why have you not changed it? 18. Because your leave one thing for another, is followed by the preposition de : father has no change. 19. Have you the change for a guinea ? changer d'habit, de chapeau, etc., to put on another coat, hat, 20. No, Sir, I have only twelve shillings. etc.; changer d'avis, to change one's mind; changer de maison, SECTION LVIII.—RULES FOR THE PLURAL OF COMPOUND to move, to change houses ; changer de place, changer de pays,

NOUNS. changer de climat, to go to another place, country, climate ; changer de nom, to change one's name. The student will per

1. We have given, in Section IX., rules for forming the plural ceive that the noun following changer is not preceded by a of nouns, but, in accordance with our plan of not presenting too possessive adjective, like the noun of the English sentence. many difficulties at once, we have deferred until the present Voulez-vous changer d'habit? Will you change your coat ?

section the rules for the formation of the plural of compound Ce monsieur a changé de nom, That gentleman has changed his name.

2. When a noun is composed of two substantives, or of a 2. Changer contre means to exchange for ; changer pour, to substantive and an adjective, both take the form of the plural: change for, to get change for.

un chef-lieu, des chefs-lieux, a chief place, chief places; un Voulez-vous changer votre chapeau Will you exchange your hat for gentilhomme, des gentilshommes, a nobleman, noblemen ($ 9 contre le mien ?


(1) (3)]. Changez ce billet pour de l'argent Change that noto for silver.

3. When, however, two nouns are connected by a preposition, 3. Tarder means to tarry, to be long in coming. Tarder used the first only becomes plural: un chef-d'œuvre, des chefsunipersonally, and preceded by an indirect object, means to long, d'ouvre, a master-piece, master-pieces [§ 9 (2)]. to wish for.

4. In words composed of a noun and a verb, preposition or Votre sæur tarde bien à venir, Your sister is very long coming.

adverb, the noun only becomes plural: passe-port, passe-ports, Il me tarde de la voir, I long to see her.

passport, passports (S 9 (6)].

5. Words composed of two verbs, or of a verb, and adverb, RÉSUMÉ OF EXAMPLES.

and a preposition, are invariable: un passe-partout, des passeN'avez-vous pas changé d'apparte. Have you not taken another apart. partout, master-key, master-keys ($ 9 (8)]. ment? ment?

6. We have seen [Sect. III. 4] that the name of the material Nous avons changé de maisons. We have changed houses.

always follows the name of the object, and that both are united Votre frère a changé de conduite. Your brother has changed his con- by the preposition de. The name of the profession or occupa

duct. Contre quoi avez-vous changé votre For what have you exchanged your

tion also follows the noun representing the individual, and the

same preposition de connects the two: un maître d'armes, a cheval ?

horse ? J'ai besoin de monnaie, pouvez. I want change, can you change me fencing-master; un maître de dessin, a drawing-master; un vous me changer cette pièce de this twenty-franc piece ?

marchand de farine, a dealer in flour ($ 76 (12), $ 81 (4)]. vingt francs ?

7. The name of a vehicle, boat, mill, etc., always precedes tha Ce garçon a beaucoup tardé. That boy tarried very much noun describing the power by which it is impelled, or the purIl nous tardait d'arriver. We longed to arrive.

pose to which it is adapted; the name of an apartment, that of Il leur tardait de revoir leurs amis. They longed to see their friends the use to which it is appropriated. The connecting preposition

again. Il me tarde de revoir la France.

is à : un moulin à vapeur, a steam mill; un bateau à vapeur, a I long to sec France again,

steamboat ; un moulin à eau, a water-mill; la salle à manger, VOCABULARY.

the dining-room ($ 76 (13), § 81 (5)].
Air, m.,
Gris, -e, grey,
Pagsé, -e, past, last.

Avis, m., mind, mean-Guinée, f., guinea. Pays, in., country.
Jeune, young.
Rentr-er, 1, to come in

Lille et Arras sont les chefs-lieux Lisle and Arras are the chief places Blanc, .che, white. Maitre, m., master. again.

des départements du Nord et du of the departments of the North Chang-er, 1, to change, Manteau, m., cloak. Schelling, m., shilling.


and of the Pas-de-Calais. to alter. Monnaie, f., change. Vie, f., life, conduct.

Les chemins de fer et les bateaux Railroads and steamboats are very Combat, m., combat. Mouillé, -e, wet. Visage, m., counte

à vapeur sont très-nombreux en numerous in America, Conduite, f., conduct. Parceque, because.

Amérique. nance, face.

Cette maison contient un salon, That house contains a drawing-roos, EXERCISE 109.

une salle à manger, une cuisine, a dining-room, a kitchen, and 3*** 1. Cet homme n'a-t-il pas changé de vie ? 2. Il a changé Les moulins à vent sont plus com- Windmills are

et plusieurs chambres à coucher. ral bedrooms. de conduite. 3. Cette grande maison n'a-t-elle pas changé de

muns en France que les moulins France than water or stram-mills, maitre ? 4. Elle a changé de maître, le Capitaine G. vient de

à eau ou à vapeur. l'acheter. 5. Vous êtes mouillé, pourquoi ne changez-vous

VOCABULARY. pas de manteau ? 6. Parceque je n'en ai pas d'autre. 7. Votre cousine ne change-t-elle pas souvent d'avis ? 8. Elle en change Båt-ir, 2, to build.

Armes, f. pl., fencing. Dessin, m., drawing. bien souvent. 9. Pendant le combat, ce jeune soldat n'a-t-il Bouteille, 2., bottle.

Engag-er, 1, to engage. pas changé de visage ? 10. Il n'a point changé de visage., 11. Cabriolet, m., gig.

Faire båt-ir, 2, to have Vapeur, f. vapour, stara

built. Ce malade ne devrait-il pas changer d'air ? 12. Le médecin Chat-huant, m., owl. Se munir, 2 ret., to pro. Voiture, 2., carria, lui recommande de changer de pays. 13. Où est votre cheval Chauve-souris, f., bat. vide one's self with. gris ? 14. Je ne l'ai plus, je l'ai changé contre un blanc. 15. Avec qui l'avez-vous changé ? 16. Je l'ai changé avec le jeune

EXERCISE 111. homme qui demeurait ici le mois passé. 17. Le marchand 1. Faut-il avoir un passe-port pour voyager en France ? ? peut-il me changer cette pièce de quarante francs ? 18. Il ne n faut en avoir un. 3. Les Anglais se munissent-ils de passe saurait (cannot) vous la changer, il n'a pas de monnaie. 19. ports pour voyager en Angleterre ? 4. On n'a pas besoin də Avez-vous la monnaie d'une guinée (change for a guinea) ?

passe-port en Angleterre.5. Aimez-vous à voyager sur les

chemins de fer ? 6. J'aime mieux voyager sur les chemins de EXERCISE 110.

fer que sur les chemins ordinaires. 7. Avez-vous apporté vos 1. Why do you not change your coat? 2. For a very good passe-partout? 8. Je n'ai point de passe-partout, je n'ai que reason (raison, f.), because I have no other. 3. Has your father des clefs ordinaires. 9. Votre frère est-il venu dans un batesa changed houses ?" 4. No, Sir, but we intend to do so (de le faire) à vapeur ? 10. Il est venu dans un bateau à voiles. 11. A rezto-morrow. 5. Has that child altered his conduct ? 6. He has vous une voiture à quatre chevaux ? 12. Non, Monsieur, nous altered his conduct, he is very good now (maintenant). 7. Was n'avons qu’un cabriolet à un cheval. not yor- brother afraid; did not his countenance change? 8. un moulin à vapeur ? 14. Il a fait bâtir deux moulins

, l'an Hie e changed, but he was not afraid. 9. Have you vent et l'autre à eau? 15. Votre compagnon a-t-il engagé un

chambre, f.)? 10. I have not changed maltre d'armes ? 16. Non, Monsieur, il a déjà un maitre os very good. 11. Do you not long to be in dessin et un maître de danse.

17. Combien de charabres i

more common is

Ordinaire, tusi.
Roue, f., thed.

Voile, i., sail.

Voyag-er, 1, te turels


13. Votre frère a-t-il báti coucher avez-vous ? 18. Nous en avons deux. 19. Avez-vous | 27. Cette belle petite fille n'a ni faim ni soif. 28. Qu'a-t-elle ? 29. une bouteille de vin ? 20. Non, Monsieur, mais j'ai une bouteille Elle n'a ni parents ni amis. 30. Cette montre d'or est-elle bonne ?

32. L'avez-vous ? à vin (wine-bottle) [$ 81]. 21. Voyez-vous les chats-huants ? 31. Celle-ci est bonne, mais celle-là est meilleure.

33. Je l'ai, mais je n'ai pas celle de votre scur. 34. Je n'ai ni la vôtre 22. Non, mais je vois les chauves-souris.

ni la mienne, j'ai celle de votre mère. EXERCISE 112. 1. Is your father in England ? 2. No, Sir, he is in France

MECHANICS.-XIV. with my brother. 3. Have they taken passports ? 4. Yes, Sir, they have taken two. 5. Is it necessary to have a passport to ILLUSTRATIONS OF PRECEDING PRINCIPLES-KITE, BOAT, ETC. travel in America ? 6. No, Sir, but it is necessary to have one

---ELEMENTS OF MACHINERY. to travel in Italy. 7. Is there a steamboat from Calais to We have now to trace the practical application of the principles Dover (Douvres) ? 8. There are several. 9. Is there a railroad already laid down, and the best way of doing this is to take from Paris to Brussels (Bruxelles) ? 10. There is one from some common instances and carefully examine them, and we Paris to Brussels, and one from Paris to Tours. 11. Has your shall see that the same rules will apply to other and more combrother bought a windmill ? 12. No, Sir, but he has built a plicated cases. steam-mill. 13. Are there many wind-mills in America ? 14. A heavy box is resting on a four-legged table. What are the No, Sir, but there are many water and steam-mills. 15. Does forces that act on it, and what on the table ? On the box there Four cousin learn drawing ? 16. He does not learn it, he are only two—its own weight acting downwards, and the upward rannot find a drawing-master. 17. Is the fencing-master in the pressure of the table, exactly counterbalancing this weight. We aining-room? 18. No, Sir, he is in the drawing-room. 19. Is turn, then, to the table, the forces acting on which are not quite your cousin in his bedroom? 20. No, Sir, he is out (sorti). so easily determined. There are its own weight and the weight 21. How many rooms are there in your house ? 22. Five: а of the box acting through their respective centres of gravity. kitchen, a dining-room, a drawing-room, and two bedrooms. 23. These are parallel forces, and, as we have seen, have a resultant Are there owls here ? 24. Yes, Sir, and bats too.

equal to their sum, and acting at a point in the line joining them, so taken that their distances from it are in the inverse

proportion to their intensities. The other force which acts on KEY TO EXERCISES IN LESSONS IN FRENCH.

the table is the resistance of the floor on which it rests, which EXERCISE 18 (Vol. I., page 78).

resistance is transmitted upward through the four legs. If the 1. Avez-vous mes tables ou les vôtres ? 2. Je n'ai ni les vôtres ni weight act at a point equally distant from these, each bears an les miennes, j'ai celles de l'aubergiste. 3. Les avez-vous ? 4. Non, equal share; but if not, it is divided between them in the inverse Monsieur, je ne les ai pas. 5. Votre soeur a-t-elle mes chevaux ? proportion to their distances. To make this more clear, we will 6. Oui, Monsieur, elle a vos deux chevaux et ceux de votre frère. suppose these distances to be 2, 6, 6, and 8 feet respectively. Find 7. Avez-vous raison ou tort? 8. J'ai raison, je n'ai pas tort. 9. Le the least common multiple of these numbers, that is, the least ferblantier a-t-il mes chandeliers d'argent ou les vôtres ?

10. Il n'a number each will divide without any remainder. In this case ni vos chandeliers d'argent ni les miens. 11. Qu'a-t-il? 12. Il a les it is 24. This we divide successively by the distances, and obtain tables de bois de l'ébéniste. 13. A-t-il vos chaises d'acajou? 14.

Non, the quotients 12, 4, 4, and 3; and these numbers represent the Monsieur, il a mes tables de marbre blanc. ci ou celles-là? 16. Je n'ai ni celles-ci ni celles-là, j'ai celles de proportion in which the weight is divided between the legs. l'ébéniste. 17. Avez-vous de bons porte-crayons ? 18. Non, Monsieur,

Now suppose the weight of the table and box to be 207 pounds. mais j'ai de bons crayons. 19. Lo voyageur a-t-il des fusils de fer Since 12, 4, 4, and 3, added together make 23, the log 2 feet off 20. Oui, Monsieur, il a les miens, les vôtres et les siens. 21. N'a-t-il pas supports 12 parts out of every 23, i.e., 1 of the weight, or 108 ceux de votre frère ? 22. Il n'a pas ceux de mon frère. 23. L'ouvrier pounds. Those 6 feet off support is or 36 pounds each; and the a-t-il nes marteaux de fer? 24. Oui, Monsieur, il les a. 25. Mon other sustains only is or 27 pounds. A calculation of this sort is trère a-t-il vos plumes ou celles de mon cousin ? 26. Il a les miennes et les vôtres. 27. Avez-vous les habits des enfants ? 28. Oui, Madame, mine the relative strength the dif

very frequently required to deter

RN je les ai. 29. Avez-vous le chapeau de votre sour? 30. J'ai celui de ma cousine. 31. Votre frère a-t-il quelque chose ? 32. Il a froid et ferent parts of a building should faim. 33. Avez-vous des chevaux ? 34. Oui, Monsieur, j'ai deux have. chevaux. 35. J'ai deux matelas de crin et un matelas de laine.

We will take another case. A

body, G (Fig. 86), rests on an in- FZ
EXERCISE 19 (Vol. I., page 79).
clined plane, the angle of which, A.

В 1. Is that lady pleased ? 2. No, Sir, that lady is not pleased. 3. Is CA B, is 30°, and the co-efficient of your daughter quick? 4. My son is very quick, and my daughter is friction is 4. What forces act on idle. 5. Is she not wrong? 6. She is not right. 7. Is your cousin G, what are their amount, and what happy? 8. Yes, Madam, she is good, beautiful, and happy. 9. Has other force must be applied to keep she friends! 10. Yes, Sir, she has relations and friends. il. Has she it in its place? We will try and a new dress and old shoes ? 12. She has old shoes and an old dress.

Fig. 86. 13. Has not your brother & handsome coat? 14. He has a handsome solve these questions. As already coat and a good cravat. 15. Have you good meat, Sir? 16. I have seen, three forces act on 6-its weight acting along Gw, the resistexcellent meat. 17. Is this meat better than that? 18. This is better ance of the plane acting along G R,

and the force of friction, which than that. 19. Has your friend the beautiful china inkstand ? 20. His is of the weight and acts along FG. We found, when considerinkstand is beautiful, but it is not china. 21. Is any one hungry? ing the inclined plane, that the power necessary to sustain a 22. No one is hungry. 23. Are the generals here? 24. The generals must bear the same proportion to its weight that Bc does to A C. and the blacksmiths are here. 25. I have your parasols and your This, then, is the first thing we must find out, and we must umbrellas, and your children's.

have a slight acquaintance with mathematics for this; there is, EXERCISE 20 (Vol. I., page 79).

however, no difficulty in the matter. Produce C B to D, so as

to make BD the same length as B C, and join A D. The triangles 1 votre petite seur est-elle contente ? 2. Oui, Madame, elle est contente.

ABC and ABD are exactly equal. For as Bc is equal to BD, 3. Cette petite fille est-elle belle ? 4. Cette petite fille n'est pas belle, mais elle est bonne. 5. Avez-vous de bon drap et de and each is at right angles with A B, A B C would, if we were to bonne soie ? 6. Mon drap et ma soie sont ici.

7. Votre scur est turn it over, exactly lie on A B D. AD is, then, equal to AC. Now elle heureuse ? 8. Ma soeur est bonne et heureuse. 9. La seur de ce in any and every triangle the three interior angles are together médecin a-t-elle des amis ? 10. Non, Madame, elle n'a pas d'amis. equal to 180°, or two right angles, and in A B C we know that 11, Votre viande est-elle bonne ? 12. Ma viande est bonne, mais mon the angle CAB is 30°, and A B C, being a right angle, is 90°; fromage est meilleur. 13. Le libraire a-t-il un bel encrier de porce- therefore A C B must be 60°, and A D B is equal to it, and therelaine? 14. Il a un bel encrier d'argent et une paire de souliers de cuir. fore is also 60°. The angle BAD is likewise equal to BAC, and 15. Avez-vous mes parasols de soie ? 16. J'ai vos parapluies de coton. 17. L'habit de votre frère est-il beau ? 18. Mon frère a un bel habit the angles of the triangle cad is 60°, and therefore they are

as each is 30°, the angle CAD is 60°. We see thus that each of et une vieille cravate de soie. 19. Avez-vous des parents et des amis ? 2). Je n'ai pas de parents, mais j'ai des amis. 21. Cette belle dame equal to one another; and, since the angles are equal, the sidos a-t-elle tort? 22. Cette belle dame n'a pas tort. 23. Avez-vous de

are also equal, for there is no reason why one should be greater belle porcelaine ? 24. Notre porcelaine est belle et bonne. 25. Elle than another. The triangle is thus equilateral and equiangular. est meilleure que la vôtro. 26. Cette petite fille n'a-t-elle pas faim ? | We have now found out what we wanted; for, if Bo be repre


sented in length by 1, CD will be 2; and Ac is equal to CD, Here is another case, involving the same principle. A bracket therefore it is also 2, and the proportion bc bears to A c is 1 A B (Fig. 88), projects from a wall, to which it is fastened by a tá 2. or On an incline of 30°, then, the power must be half screw. A strut, A C, supports the outer end, and a weight, w, the weight; but, in this case, friction sustains one-fourth, and rests on it. In what direction is the strain on the screw? The therefore a power must be applied, acting in the direction GP, three forces here are, gravity acting along w g, the thrust of and equal to one-fourth of the weight, in order to maintain the beam, A c, and the strain on the screw. The two former equilibrium.

act through the point o, and the direcWe have now to solve the remainder of the question. We tion of the third is found by drawing a know how much Gw, G F, and GP are ; but we want to know line from this point to the scrow. We what portion of the weight is borne by the plane, that is, what may take a line, oc, of such a length

B proportion A B, which represents the resistance of the plane, as to represent the weight, and resolve bears to AC. To find this, we need another very important it into o b and o a, one acting along geometrical proposition, which you will find fully proved in the strut, the other perpendicular to Lessons in Geometry, Problem XXX., Vol. I., page 337.

the wall. These will represent two In every right-angled triangle the square described on the forces, which are together equivalent to side opposite the right angle is equal to the sum of the squares o c, and of these o b will be overcome on the sides containing it. If we represent the sides by numbers by the pressure of the strut, and the expressing their lengths, the rule holds equally true. Suppose, other force, o a, tends to draw the for example, wo measure off from one of the sides containing screw from the wall. A portion, how. the right anglo a length of 4 inches, and from the other a length ever, of the pressure of the strut will of 3 inches, we shall find the line joining these two points is be borne by the screw, and these two equal to 5 inches.

forces combining produce the resultant, The square of 4 (which means 4 multiplied by itself) is 16, which acts on the screw towards the

Fig. 88. and the square of 3 is 9. These added together make 25, which point o. is the square of 5. The usual way of writing this is 4*+32 = 5. It is frequently very important to be able thus to tell in In this way, if we know the length of any two sides of a right- what direction a strain will act, as the strength of our materials angled triangle, we can always calculate the third. Now in the must be proportionate to it. In this way the direction of the case we are examining, we know that the side a c is equal to 2 tie-beams and king and queen posts of a roof are determined. and the side Bc to 1; but the square of Ac is equal to the sum We know, too, where to apply struts and braces to the of the squares A B and BC; the square of AB must, therefore, be framework of a building, so as to gain the greatest benefit from equal to the difference between those of AC and Bc. Now these them. are 4 and 1; the square of A B is, then, equal to 3, and the length Now there are one or two cases of the composition and resoluof A B must be represented by the quantity which, multiplied by tion of forces that are frequently given as illustrations, and if itself, will make 3. This is called the square root of 3, and is we clearly understand them we shall be able to master most written ✓3. By arithmetic we can easily find exactly what this others. The first is that of a kite, for there is science to be number is, but you can see that it is very nearly 11. The pro- learned even from this common plaything. Indeed, we shall portion, then, of A B to Ac is 11 to 2, or 7 to 8, and the plane frequently find, among the very commonest things, good illustrasustains a pressure equal to about of the weight. We have tions of any subject we may be studying. thus discovered the magnitude of all the forces as required. Let < (Fig. 89) represent a kite. The forces which act on it are,

When, as in our last lesson, we have resolved all the forces the force of the wind, acting, we will suppose, in the direction of acting on a body along two lines at right angles, we can in this the arrow, the tension of the string, acting along it in the direzway find the magnitude of the resultant without the trouble tion K s, and the weight of the kite; and by the action of these and possible inaccuracy of actual measurement. Suppose we three forces it is kept at rest. We will consider them singls; have a remainder of 12 pounds acting along one of the lines, and first, we will take the force of the wind. Take a w, of such and one of 5 pounds along the other, the resultant will be equal a length as to represent this force. The kite is always so made to V129 + 5; that is, to the square root of 144 + 25, or 169, as to present a large surface to the wind in proportion to its which is 13. In the same way we can solve many questions weight, and the string is fastened to the loop in such a way that frequently met with. Here is an example. Two forces act on it does not hang vertical, but inclined at an angle; the tail

, a body; the resultant is 34 pounds, and one of the forces is 16 however, prevents its being so acted on by the wind as to come pounds; what is the other? We first find the square of 34, which is 1,156 ; from this we take the square of 16, or 256, and we have left 900. The square root of this is 30, and this accordingly is the intensity of the other force.

Now turn to another common thing. A ladder, A B (Fig. 87), leans against a wall. What are the forces acting on it? Its own weight acta vertically downwards through G, and the other forces whách keep it at rest are the reaction of the wall and the ground St A sad 3 respectively. Now there is but little friction at a, sad we say therefore consider the reaction to be in the

direction A P perpendicular to the wall.
GW and AP, then, represent two of the
forces acting on the ladder; but, as it is
at rest, all three foroes must act through the
same point. Now the only point in which wg

Fig. 89.
and A P meet is that found by producing wg
till it cuts A P. Let e be this point ; the force in the same straight line with the cord. Let us, then, resolre
at B must act in the direction B R. This force the force of the wind into two, one acting edgewise on the kite

, is the resultant of two others—the resistance the other perpendicular to its surface. We draw the parallelo of the ground acting vertically upwards, and gram Kowa, and thus have the two forces, x 0 and 6 A, instead the force of friction which acts along the of kw. K 0 has no effect, as it acts on the edge, and we need,

ground and towards the base of the wall; and therefore, only consider the part K A. Fig. 87. we easily see that the more nearly vertical the Now we will introduce a second force, that of the string.

ladder is, the greater is the former as com. Produce s k backward, draw A B perpendicular to B K, and comme parod with the latter, and therefore the less the amount of plete the parallelogram D K BA. friotion which is required to keep it in its place. We have thus into KD and K B. The latter will be expended in stretching the

We can again resolve & A the noientific ronson for the well-known fact that the more a string, and have no tendency to move the kite, and thus we have Inddor in inolinod, the greater need there is for the foot to be Kd left as the effective resultant of these two forces. We con 40 as to koep it from slipping.

consider the third, which is the weight of the kite. Draw ko




[ocr errors]
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of such a length as to represent this, and complete the parallelo- Sometimes toothed wheels are used instead of straps, espegram DCGK. We have then k c the resultant of K D and KG, cially when the distance through which the power has to be and therefore of all the forces which act on the kite, and this transmitted is small. The advantage is that they do not slip, is the direction in which the kite will move, but as it does so, as straps are liable to do; the friction with them is, however, the angle at which it is inclined varies till k D and K G become greater. If we want to transmit motion from a shaft to another opposite and equal, and then the kite will remain at rest as long placed at an angle with it (Fig. 91), we employ what are known as the force of the wind remains unaltered.

as bevelled wheels. The action of these will be clear from the The other case we will consider is that of a ship, which will figure, without any explanation. sail within a few points of the wind. Letco (Fig. 90) represent Often it is required to change a rotating motion into a prothe direction and intensity of the wind, and s v the direction in gressive one, and we can accomplish this by means of a rack and

which it is desired that the vessel pinion. A number of notches are
should advance. The sail is placed cut in a bar of metal (Fig. 92a),

in the direction A B, which is mid- of such a size and at such dis-
B way between that of the wind and tances that the teeth of the

that of the vessel. We must, as wheel exactly fit into them, and
before, resolve c o into two forces, as the wheel is turned the rack
E O and ro. The part E o, which is moved onwards. This is very
acts along the direction of the frequently employed when a

sail, has no effect in moving the slow and regular motion is re-
Fig. 90.

vessel; F 0, which acts perpendi- quired, as in the adjustment of
cularly to the sail, is the effective the tube of a microscope. In-

Fig. 92. portion. We must now again resolve this force along two direc- stead of a rack a chain is sometions, one being that in which the boat moves, the other at right times used (Fig. 92 b), the links being made of a peculiar shape, angles to it. We make G o equal to F o, and about it describe so that the teeth of the wheel may catch in them. the parallelogram HG , and thus have two forces, represented The crank (Fig. 93) is, perhaps, one of the most common of these by o H and o 1, in the place of the original force, co. Now, of elements of machinery. The piston-rod of an engine is usually these, o 1 has no tendency to cause the vessel to advance; it jointed, and the end of the jointed part fixed to an arm proacts sideways on the vessel, usually inclining it, and causing a jecting from the axle to be turned, and called a crank. Someslight motion, but it is resisted by the pressure of the water times it is fixed to a pin in one of the spokes of the wheel, but against the side ; the other portion, o , represents the portion the action is just the same. The force, however, with which it of the force of the wind which is effective, and produces motion. drives the wheel is continually varying. When the piston is

In the same way you can calculate what portion of the force at the bottom of the cylinder the crank and piston-rod are in of the wind is effective in turning a windmill. The vanes are one straight line, and therefore all the power presses on the always set at an inclination with the plane in which they turn, bearing of the axle, and is lost. As the wheel turns the power and you must resolve the force along two directions, one per-acts with a loverage which increases pendicular to the surface, and the other along it. The former till the wheel has made nearly one

you again resolve, and thus find what part of it produces rota- fourth of a revolution; it is then at
tion, and what part presses against the face of the mill.

its maximum, and diminishes till the
piston-rod reaches its highest point,

Fig. 93.

when it is all again lost. Now it is As it is our object to make these lessons as practical as pos- clear that unless we have some means of regulating the speed sible, it will be well to look at a few of the simpler modes of the machine will work very unevenly, and at times stop altering and transmitting power. Sometimes this is advanced altogether. To obviate this, a large and heavy wheel, called the to the rank of a separate science, and called kinematics, or the fly-wheel, is fixed on the axle of the crank. This, when once science of motion, but it should be referred to here as a part of started, acquires an amount of momentum or moving force practical mechanics.

which carries it over the dead points when the power is lost. We seldom have our power available for use in the exact way On account of the weight of the wheel, its motion is but we desire. Sometimes we have an alternate motion, like that of slightly accelerated when the piston acts at its greatest loverthe piston-rod of an engine, and we want to derive from it a rota- age, but the additional force is stored up in it, and thus ensures tory or progressive motion; or we want to transmit it along a a steady motion. The heavy wheel of a foot-lathe serves predirection making some angle with its course, or to make many cisely the same purpose. The power is here applied during other alterations in its mode of action.

rather less than one-half of the revolution, but the momentum In large factories there is frequently a long shaft running then acquired carries it through the remaining part. along the building, and driven by an engine. From this it is required to drive all the machines in the place. This is accom

EXAMPLES plished by fixing wheels on the shaft, and letting endless straps

1. Forces of 9 and 12 act at right angles; what is their resultant ? pass over these and then over the driving pulleys of the machines.

2. The resultant of two forces which act at right angles is 10 pounds. The motion may often be greatly altered in this way. The strap One of the forces is 6; find the other. itself merely transmits the power, and whether there is a gain or 3. Two men, ono on each side of a stream, tow a barge. The angle lose in speed or power depends on the comparative size of the the two ropes make is 60°, and each pulls with a force of 100 pounds. sheaves.

Frequently there are several of these wheels of What is the total force exerted on the barge ? different sizes fixed on the axle and on the machine, and thus

4. The tension of a wire in a piano is 100 pounds, its length is 5 feet. the speed may be altered at pleasure. If the strap passes over

What force is required to draw its middle point 2 inches out of its a large one on the shaft and a small one on

position ? the machine, there will be an increase of 300. The co-efficient of friction is 1. What force is required to keep

5. A weight of 90 pounds rests on a plane inclined at an angle of speed, and if we reverse the condition there it at rest? will be a loss. A common illustration of a similar arrangement is seen in a watch. The spring when fully wound up exerts a

ANSWERS TO EXAMPLES IN LESSON XIII. much greater power than it does when the 1. The forces acting at the longer end are the power of 10 pounds watch has run nearly down. Now this acting at a distance from the fulcrum of 6: feet, and the weight of the would make it go irregularly, and therefore lever, which is also 10 pounds, and acts through its middle point, or the fusee is introduced. When the force 2 feet from the fulcrum. The moments on this end are thus 10x 67, of the spring is greatest the chain acts

or 67), and 10x21 or 27. These make 95 pounds. As w acts at a Fig. 91. on the smallest part of the fusee, and there- distance of 14 feet, it must be


or 76 pounds. fore has only a short leverage, but as it 2. Since is lost by friction, we may regard the weight as 12 pounds oxwinds and loses its force the chain acts at a greater lever only. Now of the weight of the first pulley is supported by the age, and & uniform rate of motion is thus maintained.

power, of the next, and, and is of the other two; and since each 3. 8 cwt. 1 qr. 6 lbs. 8. 19 qrs. 6 bush. 2' 14. 250 $ 15". take the first quantity for a numerator, and the latter for a


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weighs 2 pounds, these amounts are 1 pound, 1 pound, 1 pound, and 6. EXAMPLE.-Reduce £2 6s. 8d. to the fraction of £3 56. pound, together 1. Take this from 12, and we have an effective

8d. power of 10; remaining; and as the gain is 16, the weight raised is

2} pounds.

5s. od. 16 x 101, or 162 pounds.

31 pounds. 3. Friction requires a strain of 9x 20, or 180 pounds, to overcome it, Therefore the required fraction is , which is } * 18, i.e., II. and do of the weight has to be borne. The strain, therefore, is 190 + 448, or 628 pounds.

7. The question of Art. 2 might also have been solved as 4. Friction is here s' of 27 cwt., which equals % cwt. The amount follows by the aid of the present method :of the weight sustained by the horse is to of 27 cwt., or # cwt. The

25 total strain is thus 17 cwt., or 144 pounds.

25 minutes = of an hour = of a day.

60 x 24 5. The weight of the carriage is 25 * 80, or 2,000 pounds.

4 hours is of a day I of a day. 6. The co-efficient of friction is e, or nearly is.

Therefore, days 4 hours 25 minutes are 3 + + days,

60 x 24

or 3-7 of a day, or of a day. LESSONS IN ARITHMETIC.-XXVIII.

Therefore, 1 of 3 days 4 hours 25 minutes * * !! of a day.


1of a day = 1!! * 24 hours 9 hours = 61% hours. QUANTITIES.

# of an hour = }} * 60 minutes = 34 minutes. 1. To find the Value of any Fraction of a Compound Quantity.

Hence the required answer is 1 day 6 hours 34 minutes. It is evident that we have only to divide the given compound

EXERCISE 47. quantity by the denominator, and then multiply by the nume Examples in finding a Fractional Part of a Compound Quantity, rator. The first part of the process determines the magnitude

and in reducing One Quantity to the Fraction of another of of the equal parts into which the denominator indicates that the

the same kind. quantity is to be divided, and the latter takes as many of those parts as are indicated by the numerator.

Find the value, expressed in successive denominations, of Thus, of £1 = 3 x g. = 15s.

1. { of el; } of £1; I of ls. Again, it of £2 = is of 40s. = 4 * 198.

2. of £1; of ls.; 11 of 3s. 20.
Ys. = 10 s.

3. of 1 lb. avoir. ; 1 of 1 oz. Troy; 1 of 1 cwt. Again, I of a shilling is į of 12 pence, or 2 * Yd.

4. of 1 ton; of 1 yard; i of 1 rod. Therefore, it of £2 = 108. 8d.

5. of 1 mile; 1 of 1 gallon ; & of 1 peck.

6. ; of 1 hour; f of 1 minute; of 1 degree. 2. Find 1 of 3 days 4 hours 25 minutes.

7. 34 of f of a mile ; f of ff of a week.
days. hrs. min.

42 of a guinea;

of a crown.


9. If 1% of a sum of money of 5s. 100., find it. 72 + 4 = 76 hrs.

24 + 13 10. Find the value of

of £20.

11. Find the sum of of 28. 6. + of £3 2s. 6 d. + } 4560 + 25 = 4585 minutes.

of £5 7s. 31d. Therefore, of 3 days 4 hours 25 min. = of 4585 minutes 12. And of of 16s. 6 d. + } of 12s. 10 d. +of L2 48. 8.d. = 2 x 917 = 1834 minutes.

13. Find the value of of £7 14s. 1d. - of £4 Os. 1d.

14. Find the value of ji of £4 (s. 1d. - o of £3 10s. 1d. Reducing 1834 minutes to higher denominations,

15. Find the value of

| of 6,0 ) 183,4


of £1.

of g + of 24 ) 30 34 minutes.

Reduce to the fraction of a pound

16. 4 s. ; 4s. 70. ; 9s. 2 d. 6 hours.

17. 138. O d. ; 3;d. ; £3 15s. 9 d. Therefore, of 3 days 4 hrs. 25 min. is 1 day 6 hrs. 34 min.

18. What part of £1 is of a penny ? 3. To reduce one Compound Quantity to a Fraction of any

19. What part of 1 lb Troy is 7 oz. ?

20. Reduce of a quart to the fraction of a gallon. other.

21. Reduce of 1 secon to the fraetion of a week. Finding what fractional part of one compound quantity 22. Reduce £3 178, 11 d. to the fraction of £7 35. another given compound quantity is, is called reducing the latter 23. Reduce 3 pecks 2 gallons to the fraction of 2 bushels. quantity to the fraction of the first.

24. Reduce 15 cwt. 65 lbs. to the fraction of 2 tons 3 cwt. Thus, finding what fraction of one pound 6s. is, is reducing

25. Reduce 1° 15' 10" to the fraction of a right angle. 6s. to the fraction of a pound.

26. Reduce 1} acre to the fraction of 5 acres 2 r. 40 p. This is, in fact, only another name for performing the opera

27. Reduce aty of £1 to the fraction of a penny.

28. Reduce it of a week to the fraction of a minute. tion of dividing one compound quantity by another, or of finding

29. Reduce of of £2 8s. 9d. to the fraction of £1 ls. 8d. the ratio of two compound quantities (see Art. 11, Lesson

30. Express as a fraction of £10 the difference betwen £8) XXVII, page 101).

and of £8. 4. EXAMPLE.-What fraction of £1 7s. 6d. is 39. 60. ?

1 lb. 7 oz. 4 dwts. 31. Find the value of

of 15 guineas. £1 75. 60. = 330 pence.

2 lb. 7 oz. 10 dwts.

1° 17'

77 dys. 4 hrs. 30 min. 38. 6. =

32. Find the value of

of Hence, it 41 7s. 6a. be divided into 330 equal parts, 3s. 68.

6 dys. 12 hrs.

17 15

136 gals. 2 qts. of 517 square feet 72 inches. contains 42 of them; Therefore, 3s. 68. is 33% of £1 7s, 6d.; and 11% = it when

178 gals. 3 qts. reduced to its lowest terms.

33. Find the value of

£3 18s. 8d.

of 104 yards 9 inches.

£6 12s. 9d. This might have been more shortly performed as follows:£1 7s. 63. = 55 sixpences.

KEY TO EXERCISES 44, 45, 46, LESSONS XXVI., XXVIL 38. 6. = 7 sixpences. Therefore, 39. 60. is s'i of £1 7s. 60.

(Vol. II. pages 78, 102.) 5. Hence the following

EXERCISE 44. Rule for reducing one Compound Quantity to the Fraction of 1. £9 28. 8fa. 7. 35 bushels 2 pks 13. 398 cubic feet 1729 another.

2. £3 58. 104.

cubic inches, Having reduced them both to any the same denomination,




42 pence.


| 15. 35° S 30". denominator.

4. 24 tons 1 cwt. 83 lbs.

9. 55 yards 2 grs. 3, 16. 54 yrs. 2 mo 2 sks The denorion to which the quantities are to be reduced

5. 19 miles 289 rods nails.

2 feet. ia a que Sometimes we can much simplify our

10. 44 yds. 1 qr. 3 nls.

6. 1 league 1 m. 7 fur. 11. 85 acres 119 rods. 17. 2 calcul ne rather than another.

10 rods 12 feet. 12. 234 acres 138 rods. 18. 1 year 164 days.

6 qts.

10 oz.

6 ds. 2 hrs. 45 min.

6 secs.

years 88 days.

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