KEY TO EXERCISES IN LESSONS IN GERMAN. EXERCISE 181 (Vol. III., page 379). 1. Ich wünsche Ihnen einen guten Abend. 2. Ich habe das Vergnügen, Ihnen einen guten Morgen zu wünschen. 3. In der Fremde gedenken wir oft mit Liebe unserer Freunde in der Heimath. 4. Ich gebenke nächsten Monat nach dem Festlande zu gehen. 5. Gedenken Sie lange dort zu blei ben? 6. Nein, ich gedenke nicht lange dort zu bleiben, ich werte bald zurückkehren. 7. Er versuchte seinen Freund bei der Erlernung der beut. schen Sprache einzuholen, aber er konnte es nicht, denn sein Freund war zu weit vorgerückt. 8. Gedenken Sie, Ihren Bruder auf seiner Reise einzu. holen? 9. Ich holte meinen Bruder nach einer dreitägigen Reise ein. 10. Vor sechs Monaten war ich im Begriffe, nach Amerika zu gehen; nun aber bin ich sehr froh, daß ich in der Heimath geblieben bin. EXERCISE 182 (Vol. III., page 379). 5. In EXERCISE 4. 1. Given a = 2. This case appears in two forms- } (A + B) = (180° C) = 90° – C. (A + B) × (a - b) a+b C) + log. (a - b) There being a logarithmic ratio on each side of this equation, of like sign, there is no occasion to allow for the added tens, which balance each other. We have now obtained (A + B) and (A B), the sum and difference of which, by the well-known rule, give the values of A and B respectively. a. sin. C By (65), we have c= .. log. c = = log. a + log. sin. C. log. sin. A. Here again, owing to difference of sign, the "added tens balance each other. sides, a and b, and an angle, A, not (This is called "the ambiguous case.") 1. When Rudolph of Hapsburg had become emperor of Germany, the internal dissensions and the so-called club-law ceased in this empire. 2. After they had killed a few stags, they desisted from hunting. 3. It ceases raining, and we now can continue our journey 4. My brother is at home; he has already been a week in bed. Germany there are other manners and customs than in America. The imperial diets were held at Ratisbon in later years. 7. The highschool at Breslau is among the best in Germany. 8. They were just dining as we arrived there. 9. They were not accustomed to take their supper until they had done all their day's work. 10. They took their dinner in summer during fine weather under a linden-tree, which stood in the yard. 11. When the cholera raged in Paris, thousands upon thousands died of it. 12. The soldiers take the field. 13. In the last storm several ships sank. 14. The beggar goes from door to door, and from village to village. 15. This redounds to my honour, to his disgrace. 16. You might do it for my gratification. 17. The 18. That is too good for enemy steers with all sails towards the east. him. 19. I am only too certain that it will happen so. 20. That may be done too when we have first regulated our own affairs. 21. Friend, life is an earnest business; suffer its hardships: thus only will the voyage be easy to you. 22. Finally, thou landest, after all, safely on shore in thy harbour; it is called the grave. 23. He has ruined his own and his friends' fortune. 24. He has ruined his health by these labours. 25. Nelson destroyed the French fleet. 26. If he for log. sin. B always admits of two values of B (except when is not careful, his whole business may be ruined in a short time. A and B being now known, C of course is known also. Familiarity with the use of logarithms is necessarily assumed in the student, who will remember that, as 10 is added to all logarithms of trigonometrical ratios (to avoid the necessity of entering negative indices in the tables, which would otherwise arise from the fact that many of the ratios are less than unity), it is also necessary to deduct 10 from them before using them in calculations, or (what is the same thing) to add 10 to the other side of any equation in which they may appear. This has been done above. The use of logarithms is fully explained in our "Lessons in Logarithms" in the POPULAR EDUCATOR. Secondly, given two included between them. Find B, C, and c. = sin. B b We find B readily from (65); viz., whence log. sin. B = log. sin. A + log. b - log. a; and C 180° - (A + B), e is found from (65), as in the last example. Now since sin. B is also sin. (180°- B), the above equation B = 90°), one greater and the other less than a right angle; and other data have to be considered in determining which is the correct one; thus (a) If the given angle A is a right angle, or greater than a right angle, B must be less than 90°, and no doubt arises. = (B) Again, if A, though less than a right angle, together with the greater value of B, be not less than 180°, it is clear the less value of B must be adopted. For instance, if A = 80° and B = 70° or 110° (i.e., 180° 70°), it is plain that 110° is an inadmissible value for B; consequently, B = 70° and C = 30°. B, be less than 180°, it is plain that the data given apply to (7) But if A, together with the greater of the two values of 80° and B 85° or 95°, we may 1. Given a 218, b = 156, and C 38° 21′ 20′′. Find A, B, and c. 2. Given a = 5324, b 31-27, and C = 126° 36′ 6′′. Find A, B, and a 3. Given b= 173, c ́= 123, and A = 22° 13′ 30". Find B, C, and a. 4. Given b = 156, a = 130, A = 42° 25′. Find B, C, and a (give both solutions). to be acute). 3. Given a = 53, c = 47, C = 36° 42′ 30′′. Find A, B, and b (assume A Find A (obtuse), C, 62, and C = 33° 19'. Find A, B, and b. 6. Given a = 217, b = 199, and B = 62° 24′ 20′′. and c. 7. Given a 100, c = country may be made, and heights and distances accurately calculated, without a single actual measurement being taken, except one at starting called the base-line. (It is usual, however, to check a result here and there by actual measurement.) By choosing or marking spots or objects at convenient Find C, b, distances apart, the whole district is divided into triangles, and it is obvious that a knowledge of one side of the first triangle calculated (the base-line before mentioned), an instrument for measuring angles, and a level, are all that are required to enable it to be completely surveyed. log. sin. A. Similarly, log. c = log. a + log. sin. Clog. sin. A. EXERCISE 6. 1. Given a = 217, B = 56° 21′ 30′′, C = 62° 41′ 20′′. Find A, b, and c. 2. Given a = 1000, B = 120° 15′ 15′′, C = 36' 52. Find A, b, and c. XXII. To find the Area of a Triangle.-There are two useful formulæ for finding the area-one in terms of any two sides and included angle, the other in terms of the three sides. 1. Area = bc sin. A. A few cases will be given in the next lesson as illustrations of the commonest practical uses of Trigonometry, and the Key to Exercises 2 to 7 inclusive. TERMS USED IN COMMERCE.-VII. PAR OF EXCHANGE.-The comparative intrinsic value of the specie of different countries according to their fixed standards of weight and purity. PARTNERSHIP.-The combination of two or more individuals If A is a right angle, area evidently = half the rectangle of the profits, or bearing a corresponding share of the losses for the purposes of business in common, each deriving a share C under the adjacent sides, which agrees with the statement, since sin. 90° = 1. If A is acute, as in Fig. 15, drop CP perpendicular to A B, or AB produced (Fig. 16). Then, by Euclid II. 1, Area But ABC, and since = ABX CP. sin. A, C P = b sin. A; therefore area = bc sin. A. If A be obtuse (Fig. 17), drop C P as before, on B A produced; then area}ABX CP, But A B = C and CP = C A sin. CAP, and sin. CAP= sup plement of A. .. Areabc sin. A........ Or, log. 2 area = log. blog. cx log. sin. A (77) 10. 2. Substituting in (77) the value of sin. A given in (73), we get Areas (s− a) (s - b) (s c)............ (78) log. s + log. (s − a) + log. (s—b) + log. (s — c) Or log. area = 2 ; arising from it. PASS BOOK. A book passing between bankers and their customers, which records all payments and receipts. PASSPORT.-A document granted by a consul, giving a description of the owner, and entitling him to pass through or to reside for a time in the country for which it is given. In maritime law, a document carried in time of war by a vessel to prove her nationality. PATENT-LETTERS PATENT.-A privilege granted under the Crown seal, conveying to the persons specified the sole right to make use of some new invention or discovery therein stated. PENALTY. A sum to be forfeited for the non-completion of a contract or for a part of it. PER CENT.-"By the Hundred." Thus 5 per cent. would be five out of every hundred. PERMIT.-A licence from the Excise authorities permitting the removal of goods upon which duty has been paid. PILOT. A person duly qualified and authorised to conduct ships through rivers, into or out of port, or through certain channels or roads. PLANT. A trade term comprehending fixed machinery, implements, or other requisites for carrying on a business. companies and underwriters secure to the parties contracting POLICY OF INSURANCE.-A document by which insurance with them for life, fire, or marine insurance, an indemnity against loss from the risk incurred. It is a document of considerable importance; stating the names of the insurers and of the insured, the amount and exact nature of the indemnity, and of the risk incurred. POST, To (Book-keeping).—To transfer an entry from one book to another. POSTDATE.-To date a letter or document of any description but it is often easier to work out (77) arithmetically than to later than the day on which it is written. employ logarithms. 2. Given a = 1000 yards, b = 2.5 miles, and C = 42°. Find the area. 6. Given a 603, b =507, c = 721. Find the area. Find the area. 8. Given a = We have now concluded our investigation of theoretic Trigonometry, or rather of such parts of the theory as will enable us to apply our knowledge largely in practice. There are formulæ for other ratios or values, such as for the radius of the circle inscribed in, or circumscribed about, a given triangle, the area of the circumscribed circle in terms of the sides, the area of any polygon inscribed in a circle (whence the area of the circle itself may be obtained approximately), and the like; but these, although useful, are not needed to enable us merely to solve "heights and distances," upon which the practical art of surveying mainly depends. A complete survey of a coast or POST OBIT BOND.-A bond, the main condition of which is that it only becomes payable after the death of some person whose name is therein specified. PRÉCIS-WRITING.-Writing the contents of a document in as short and condensed a style as possible. PREMIUM.-An additional sum beyond a standard or fixed price. PREMIUM (INSURANCE).-The per-centage or sum paid by the insured for the indemnification granted by the insurer. PRESENTMENT OF A BILL.-The act of demanding, or presenting for, acceptance or payment. PRICE CURRENT.-A list or enumeration of various articles of commerce, with the market price of each. PRIMAGE, OR HAT MONEY.-A customary percentage paid by shippers, in addition to the freight of goods, and considered to be for the master of the vessel, for his care and trouble in taking charge of such goods while on board. PRIME COST. The first cost, before charges begin to accrue. PRINCIPAL signifies the responsible person. It also applies to the partners in any establishment, who are spoken of as the principals. In Banking, the sum on which the interest arises. PRIVATEER.-A private ship fitted out for warlike purposes § 98.-THE PRONOUN.-PLACE OF THE PERSONAL PROunder a license from the Government. (See Letters of Marque.) PROCEEDS. The actual result or sum produced by any sale. PROCTOR.-An officer in the admiralty and ecclesiastical courts, corresponding with an attorney in common law or a solicitor in equity. NOUN, SUBJECT OF THE VERB. French, as well as in English, placed before them in affir(1.) Personal pronouns, used as subjects of verbs, are in mative and negative sentences:J'inventai des couleurs, j'armai la calomnie, J'intéressai sa gloire; il trembla I invented colours, I armed calumny, I touched his glory; he trembled for his life. (2.) In affirmative or negative sentences commencing with à moins, à peine, encore, peut-être, en vain, du moins, combien, etc., the pronoun may elegantly be placed after the verb, although this construction is not imperative :Peut-être vous entretiendrai-je Perhaps I will converse with you aussi de l'astronomie, on astronomy. (3.) In exclamations, the nominative pronoun is often placed after the verb in French, as well as in English:Puissé-je de mes yeux y voir May I with my own eyes see the tomber la foudre! thunder crush it! (4.) In interrogative sentences the nominative pronoun is placed immediately after the verb in the simple tenses, and between the auxiliary and the participle in the compound: Où suis-je ? qu'ai-je fait ? que dois-je faire encore ? Where am I? what have I done? what have I yet to do? (5.) In the interrogative sentences with verbs having only one* syllable, in the first person singular of the indicative present, and with some verbs having more than one syllable, but in which that person ends with an s preceded by a consonant, the pronoun je is not placed after the verb. In such case another construction must be given to the sentence :Est-ce-que je dors ? Est-ce-que je comprends ? Do I sleep? (6.) The same construction is admissible, though not desirable, with all the persons and tenses which may be used interrogatively. § 99.-REPETITION AND OMISSION OF THE NOMINATIVE PRONOUN. (1.) It is proper to repeat the personal pronouns je, tu, il, nous, ils, before every verb :: Je lis, j'écris, je me promène, I read, write, and walk. (2.) The omission of the pronouns je, tu, il, nous, vous, ils, before the second or third verb of a sentence, is a matter of choice, and subject to the following restrictions: These pronouns must be repeated 1. When the verbs are not in the same tense : Je prétends et je prétendrai I maintain and will always maintoujours, tain. 2. When the first verb is in the negative and the second in the affirmative : Je ne plie pas et je romps, I do not bend and I break. 3. When the propositions are connected by conjunctions other than et, and; ou, or; ni, nor; mais, but :— Nous détestons les méchants, We detest the wicked, because we parceque nous les craignons, fear them. (3.) Although we would advise the student to follow the first rule of this section, particularly with regard to the pronouns je, tu, nous, vous, and thereby avoid all uncertainty, we give an example where the pronoun is after the first § 100.-PLACE OF PERSONAL PRONOUNS USED AS THE regimen, makes an exception to the above rule, as it takes REGIMEN OF VERBS. precedence of the direct regimen : If men think ill of each other, at least they do not say it to each other. (4.) The Rules 1 and 2, also the Exception 3, apply to the imperative used negatively; but Rule 1 cannot apply to the imperative used affirmatively. EXAMPLES OF THE IMPERATIVE USED NEGATIVELY. Ne nous le donnez pas [Rule 1], Ne le leur prêtez pas [Rule 2], Qu'ils ne se le disent pas [Remark 3], Do not give it to us. Do not lend it to them. (5.) When the imperative used affirmatively has two regi. mens, the pronoun régime direct precedes the régime indirect in all the persons :— Envoyez-le-moi, Speak to them. Je ra'adresse à lui, à eux, I apply to him, to them. I go to you or to them. You run to him or to her. She comes to me or to you. You think of us or of him. They think of them and of you. (5.) The pronoun used as the indirect regimen of the French, which answers to the indirect object of the English preceded by a preposition other than to, and to the genitive and ablative cases of the Latin, is always, in French, placed after the verb, and preceded by one of the prepositions-de, of; pour, for; avec, with, etc. Send it to me. Put this into thy mind: he who does evil, finds evil. KEY TO EXERCISES IN LESSONS IN FRENCH. 1. Votre montre avance-t-elle ou retarde-t-elle ? 2. Elle ne retarde pas, elle va très-bien. 3. Elle retarde de vingt-cinq minutes par jour. 4. Votre pendule avance-t-elle beaucoup? 5. Elle avance d'une heure par semaine. 6. De combien retarde la montre d'or de M. votre fils? 7. Elle retarde de beaucoup; elle retarde d'une heure en vingt-quatre heures. 8. Je l'ai avancée d'une heure. 9. Je la retarderai d'une demi-heure. 10. Votre pendule ne sonne-t-elle pas la demi-heure ? 11. Non, Monsieur; elle ne sonne que les heures. 12. Avez-vous oublié de remonter votre montre à répétition ? 13. J'ai oublié de la remonter, et elle s'est arrêtée. 14. Votre montre d'argent est-elle dérangée ? 16. 15. Elle est dérangée, il faudra la faire nettoyer. Quelle heure est-il à votre montre ? 17. Il est trois heures à ma montre; mais elle avance. 18. De combien avance-t-elle par semaine ? 19. Elle avance de plus de einq minutes par jour. 20. Votre montre est-elle juste? 21. Non, Monsieur; elle n'est par juste; elle est dérangée. 22. Votre horloge sonne-t-elle juste? 23. Elle ne sonne pas juste; la sonnerie en est dérangée. 24. Avez-vous cassé les aiguilles de votre pendule? 25. J'ai cassé la petite aiguille et le cadran. 26. L'horloge a-t-elle sonné trois heures? 27. Elle a sonné midi. 28. Elle s'est arrêtée. 29. S'arrête-t-elle tous les matins ? 30. Elle ne s'arrête pas tous les matins; elle s'arrête tous les soirs. 31. Votre montre ne s'accorde pas avec la mienne. 32. N'avez-vous pas cassé le grand ressort de la montre de votre frère ? 33. Il l'a cassé en la remontant. 34. La montre de mon frère est juste, il l'a fait nettoyer et régler. 35. La montre de ma sœur n'est pas juste; elle a besoin d'être nettoyée. EXERCISE 179 (Vol. III., Page 318). 1. Had you not dislocated your arm? 2. I had not dislocated it; I had broken it. 3. If you went to America, would you resign your situation? 4. I should be obliged to resign it. 5. Is it long since 6. He resigned a month ago. 7. Has your cousin resigned his? the enemy seized upon the city? 8. He has seized upon it. 9. Will your son behave better in future? 10. He has behaved very well during his stay in Prussia. 11. Did you expect such treatment from him? 12. I did not expect it. 13. What did you expect? 14. I expected to be treated properly. 15. Why did you laugh at him? 16. Because I could not help it. 17. If you left your inkstand here, would the peasant lay hold of it? 18. He would certainly seize it. 19. Does your partner behave well towards you? 20. He behaves well to everybody. 21. Who has set your sister's 23. Has not your father dislocated wrist? 22. Dr. G. has set it. his right arm this morning? 24. He has not dislocated it; he broke it this morning at five, and Dr. S., who was present, set it imme diately. GASSIOT'S VOLTAIC ELECTRICITY.-XV. REVOLVING STAR CASCADE-THE ELECTRIC THERE are two more experiments with the induction coil which are so beautiful that they must not be passed over without notice. The first is that known as Gassiot's revolving star. For this, three vacuum tubes are mounted on a disc (Fig. 97), so as to radiate like the spokes of a wheel; they are then made to rotate rapidly by means of a multiplying wheel. By adjusting the hammer of the coil, the rate at which the sparks succeed one another can be altered, and it will thus present the appearance of a star, the number of whose rays is three, six, nine, twelve, or, in fact, any multiple of 3. The more rapidly the sparks flow, the greater the number of rays that will be seen. The other experiment is known as the Cascade. The lower part of a beaker glass is coated with tinfoil, and it is placed on the plate of an air-pump. A bell-shaped receiver with an open top is placed over it, the opening being covered by a metal plate, through the centre of which a rod passes air-tight. This rod touches the inside of the beaker, and is connected with one pole of the coil, the other being connected with the pump-plate. As soon now as the pump is worked, and the air in the receiver becomes rarefied, a faint blue light passes from the beaker to the plate; as the vacuum becomes more perfect, the cup appears completely filled with the light, which seems to flow over the edge in a continuous stream. On reversing the commutator the stream flows upwards instead of downwards. Having now seen the modes of generating the voltaic current, and conveying it from place to place, and also the principal effects which can be produced by means of it, we must turn our attention to the most important practical application of these principles; we refer, of course, to the electric telegraph. In the present day this is in almost constant use, and its employment is rapidly becoming more general; yet it is surprising to find how little is generally understood, even by welleducated people, of the principles on which it acts. From early times some mode of rapidly conveying information from place to place has been anxiously sought for, and many different schemes have been proposed. At first, the human voice was employed, and the message passed on from mouth to mouth. It was, however, soon found that signals might be seen at a distance to which the voice could never reach. Some simple signals were usually agreed upon beforehand, and transmitted by means of motions of the hand, or by a bright light. One of the earliest practical uses of this was in announcing to the inhabitants of Palestine in ancient times the appearance of the new moon, by which their festivals were regulated. After it had been seen, and information of the fact conveyed to the Sanhedrim by trustworthy witnesses, the news was telegraphed all over the land by fire signals. A party of men would ascend a hill outside Jerusalem, and kindle there a large fire of wood, pitch, and other inflammable materials. As soon as this was seen, the neighbouring hills copied the example, and thus in a short time the news was conveyed throughout the country. angles to these wires, and at the end of each was a metal spring, which might be made to touch the conductor. At the other extremity were small balls marked with the letters of the alphabet, and under each at a trifling distance small pieces of paper were to be placed. When a word or message was to be sent, the spring marked with the first letter of it was pressed by means of a glass rod against the conductor; the wire then would at once become charged with electricity, and attract the fragments of paper at the further end, indicating thereby the letter sent. The next letter would be sent in a similar way, and so the whole message was spelled out. This letter seems to be the first germ of the telegraph, but so many improvements and alterations have been made by different men that there is no one in particular to whom we can point as its inventor. The first electric telegraph ever actually erected, by which intelligible signals could be transmitted, was constructed at Geneva, by Lesage, and was somewhat on the principle of that described in the letter referred to above. Twenty-four wires were employed, and from the further end of each a small pith ball electroscope (Fig. 98) was suspended; and as soon as any wire was excited by being brought into communication with the conductor the balls would immediately diverge. Various other Fig. 97. By this plan only a few simple messages could be conveyed. A great advance on it was made many centuries later, by the use of semaphores, or arms, somewhat like railway signals. An arbitrary code was arranged by which words might be spelt out, and messages sent by these. The stations were at a considerable distance apart, and at each were placed two men, one of whom, by means of a telescope, read the signals, while the other re-transmitted them. The process was, however, very slow and uncertain, as there was no means of calling attention to the fact of a message coming; and a slight fog served at once to interrupt all communication. A telegraph worked in this way was, however, erected between London and Dover, and continued in operation some little time. Soon after the discovery of the more simple phenomena of frictional electricity attempts were made to convey communications by means of it. As early as 1727, the electrical excitement was conveyed a distance of several hundred feet by means of a wire suspended by silk threads. The excited glass was applied to one end, and it was found that particles of paper or other light substances were attracted at the other extremity of the wire. About five-and-twenty years later, a letter appeared in the Scots' Magazine, suggesting a means of communicating with a friend at a distance by means of electricity. A number of wires, one for each letter of the alphabet, were to be taken, and supported by some non-conducting cement. The excited barrel, or, as we should now call it, the prime conductor, was placed at right attempts were made to construct telegraphs to act by means of frictional electricity, and several important improvements were made, but none were ever brought into practical operation. About the beginning of the present century several facts were discovered in connection with voltaic electricity, and attention was soon turned to that as a mode of transmitting signals. The voltaic pile was, however, at first the only means of producing a current, and this was of necessity somewhat feeble. A telegraph was constructed by Sömmering at Munich in 1808 which was worked by a pile, and in which the messages were received by the decomposition of water by the current. Thirty-five wires were employed to represent the letters and numerals; these terminated in as many gold points, placed side by side in a trough of water, and lettered. When the current was sent along any wire minute bubbles of gas were evolved at the gold point, and indicated the letter. The discovery by Oersted, in 1820, that a magnetic needle was deflected from its position by an electric current passing along a wire near to it, constituted a red-letter day in the science of telegraphy; and the discoveries of Faraday, a few years subsequently, as to the phenomena of induced currents, greatly aided in bringing the science to the degree of perfection it has now attained. Galva nometers were soon constructed for the purpose of receiving the messages, and a powerful bar-magnet was frequently used to induce the current. The great drawback in the early forms had been the number of wires required, which added very greatly to the cost, and to the difficulty of maintaining the apparatus in working order. These have been gradually reduced in number, till now the great majority of instruments require only a single one. At first a return wire was always employed to complete the circuit. Steinheil, however, in experimenting with a view of ascertaining whether the metals of a railroad could be employed as conductors, made the important discovery that the earth itself would serve the purpose of a return wire. Since then a separate wire is always dispensed with, and a metal plate is buried in the earth near each telegraph station, the return wire being connected to it. We must not pursue further the history of the science, except just to state in a few words that Mr. Cooke was the first to make an experimental application of telegraphy on the Liver pool and Manchester Railway in February, 1837, and that be and Professor Wheatstone, to whom he was then introduced, from that time turned their joint attention to the preparation of our present system of communication by means of the electric telegraph. |