SECTION LXIV.-ARALIACEE, OR IVY WORTS. Characteristics: Calyx adherent to the ovary; petals five or ten inserted upon an epigynous disc, sessile; valvular in æstivation; stamens inserted with the petals, in number equal to the latter, and alternate with them, or double their number; ovary inferior, three or more celled, cells uniovulate; ovules pendent, reflexed; styles equal in number to the cells, distinct or coherent; fruit bacciform, dry, or fleshy; seed inverse, dicotyledonous; embryo very small at the base of a fleshy albumen; radicle superior. The Araliaceae generally possess a woody stem, and have for the most part alternate and stipulate leaves. Flowers regular, capitular, umbelliferous, or in racemes. The Araliaceae are inhabitants of tropical and temperate regions of both hemispheres. This natural order has a certain resemblance to the Umbellifera, both in general aspect and chemical qualities. In Araliacea, however, four, alternate with the petals; ovary inferior, two-celled, cells uniovular; ovules pendent, reflexed; style simple; fruit baccate, two or three celled; seeds inverted; embryo dicotyledonous, straight in the axis of a fleshy albumen; radicle superior. The Cornacea are generally trees or shrubs with opposite, simple, ex-stipulate leaves; flowers disposed in a capitulum, umbel, or corymb. The Cornacea are allied with the Caprifoliacea, from which, however, they may be distinguished by their free petals. They are also allied with Araliacea, but differ in their two-celled ovary and opposite leaves. Ali The Cornaceae inhabit the temperate and cool regions of the northern hemisphere. Certain members of the order possess in their bark a peculiar bitter principle termed corniine, also an astringent matter. Some produce edible fruits and oily seeds. The greater number possess a wood of great hardness. The cornel-tree (Cornus mascula) is generally 214. THE GRAPE VINE (VITIS VINIFERA). 215. THE COMMON IVY (HEDERA HELIX). the aromatic resinous principles are masked by astringent and bitter matters. The common ivy needs no description as to general appearance. Its leaves, when bruised, are aromatic, and their juice; incorporated with fatty matter, constitutes a good application to burns. The ginseng (Panax Schinseng) grows in Tartary, China, and Nepaul. Its root contains a bitter, an acrid, and a saccharine matter. The plant enjoys in Asia an immense reputation as a tonic, and sells for three times its weight in silver. Panax quinquefolium grows in North America; its root is collected and sold to the Chinese as a substitute for the real ginseng. The Aralia nudicaulis, a North American plant, is celebrated as a sudorific, and its roots are used for the purpose of adulterating sarsaparilla. SECTION LXV.-CORNACEAE, OR CORNELS. Characteristics: Calyx adherent to the ovary; petals four, inserted upon an epigynous disc, valvate in aestivation; stamens 216. THE RACEME-FLOWERING ARALIA (ARALIA RACE MOSA). 217. THE DOGWOOD (CORNUS SANGUINEA). diffused over most parts of the world. The Cornus sanguinea (Fig. 217) produces bitter and nauseous fruits, but the seed yields an oil useful for illumination and the fabrication of soap. The Benthamia fragifera, or strawberry-fruited Benthamia, is a shrub of Nepaul and Japan, now generally cultivated in European gardens. The name fragifera is given to this plant on account of its bearing a fruit similar in general appearance to a strawberry. The Cornus florida is a North American shrub. It possesses an astringent bark, and is employed as a substitute for quinine by American medical practitioners. The Aucuba Japonica, or variegated laurel, is a Japanese evergreen, with shining, opposite, coriaceous leaves, and dioecious, small axillary flowers, disposed in panicles. Ovary adherent, unilocular, uniovulate; ovule pendent, reflexed. Fruit, a scarlet berry. This shrub, which is full of ramifications and very elegant, is a well-known garden ornament. LESSONS IN GERMAN.-XXXVIII. SECTION LXXIV.-IDIOMATIC PHRASES (continued). Nicht wahr? is it not true (literally, not true?), answers to our phrases, "isn't it?" "wasn't it?" "don't they?" etc., after an assertion; as:-Es ist kaltes Wetter, nicht wahr? it is cold weather, is it not? Sie fennen ihn, nicht wahr? you know him, do you not? Sometimes nicht wahr precedes the assertion, as :-Nicht wahr, Sie sind müde, you are tired, are you not? 1. Aufwarten (compounded of the particle auf and warten (§ 90) signifies to wait upon, to serve, and governs the dative. Ich warte Ihnen auf, I wait upon you. Darf ich Ihnen mit einer Tasse Thee aufwarten? may I serve you with a cup of tea? Ich danke Ihnen, sometimes abbreviated to Ich taufe, means in addition to our "I thank you," also, "No, I thank you," according to the signification intended to be given. Ich bin so frei (literally, I am so free) or Ich bitte, is the usual equivalent to our "if you please." 3ch mache ihm meine Aufwartung, I wait upon him (literally, make my waiting upon him). Warten, when followed by the preposition auf, signifies "to wait for;" as :-Ich warte auf ihn, I am waiting for him. 2. Sollen (see § 83 [6] Remark [13]), with an infinitive, is often answered, in English, by the infinitive only, preceded by the preposition "to; as:—Ich weiß nicht, was ich thun soll, I do not know what to do. läger ließ ten Angeklagten nicht zu Worte kommen, sondern fahr immer we had. 1. Your friend whom we saw the day before yesterday is sick, is he not? 2. It was an agreeable evening, was it not, my friend? 3. Yes, it was; and I shall never forget the pleasure 4. Your brother was also there, was he not ? 5. It is yet early, is it not? 6. No, it is very late, and we must go. 7. I have waited already an hour for my friend, but still he has not come. 8. I am waiting for our servant. 9. Do not wait for him, I have just sent him out. 10. After I arrived in London, I went directly and waited upon my friend, for whom I had letters of recommendation. 11. May I serve you with a cup of chocolate? 12. No, I thank you. 13. Will you not visit us before you go to the Continent? 14. Yes, I shall pay you a visit. 15. May I help you to a glass of ale? 16. I thank you, I never drink it. 17. I have heard the news, but I do not know what to say to it. 18. You speak French and German, do you SECTION LXXV.—IDIOMS RELATING TO VERBS. Schmerzen, to pain, is used like the corresponding English word; as:— -Der Getanke schmerzt mich, the thought pains me. Die Wunde schmerzt ihn, the wound pains him. not? 1. Web (pain), joined with thun (to do, to make), forms the phrase Weh thun, to pain, to grieve (literally, to make, or cause pain); as:-Das thut mir weh, that grieves me (it causes me pain), Die Hand that Er hat dem Kinte weh gethan, he has hurt the child. ihm weh, the hand pains him. Das Kind hat sich weh getḥan, the child has hurt itself. 2. Leid thun (literally, to make, or cause pain) is employed to denote mental sufferings; sorrow; as :— Es thut ihm leid, daß er es gethan hat, he is sorry that he has done it. Es thut mir leid, ihn nicht gesehen zu haben, I am sorry not to have seen him. 3. Fehlen, to fail, to miss, to lack, is often used impersonally; as:-Es fehlt ihm an Verstand, he was lacking in understanding. So, also, Was fehlt dem Manne? what ails the man? Was fehlt Ihnen, what ails you, or, what is the matter with you? He did not know what to do. EXERCISE 142. 1. Es war eine schöne Stunde, nicht wahr, mein Freund? 2. Ja, das war es, und nicht so bald werde ich sie vergessen. 3. Nicht wahr, der Nachbar war ebenfalls auf dem Feste? 4. Ja, er war dort, und sehr vergnügt. 5. Nicht wahr, es ist schon sehr spät? 6. Nein, es ist noch ziem, lich früh. 7. Nicht wahr, es ist nicht Alles wahr, was die Leute sagen? 8. Nein, nicht Alles darf man Ihnen glauben. 9. Ich habe schon eine Stunde auf ihn gewartet, und immer läßt er sich noch nicht sehen. 10. Wir warten auf den aufwartenden Kellner. 11. Wenn Sie es erlauben, werde ich Ihnen heute Nachmittag meine Aufwartung machen. 12. Darf ich Ihnen mit einer Tasse Thee oder Kaffee aufwarten? 13. Ich danke für Thee, aber ich bin so frei, eine Tasse Kaffee anzunehmen. 14. Bei der Krönung der deutschen Kaiser zu Aachen warteten die anwesenden Fürsten repent. Gott'losigkeit, f.wicked ness. Hinzufügen, to add to, ject, discourage, contention. Es fiel nichts von Bedeutung vor. Sagen Sie mir, was Ihnen fehlt, und was die Ursache Ihrer Thränen ist. meiner Leiten abʻzuwarten. eignen kann, thut mir weher, als humour, out of tune. Volkslied, n. national song. Vor'fallen, to happen, to come to pass. Wahl, f. choice. Vorsichtig, careful. Web. (See R. 1, above.) 3ufriedenheit, f. contentedness. Zu'fügen, to cause, to inflict. Nothing important happened. I lack patience to await the end auf. 15. Umsonst habe ich ihn darauf aufmerksam gemacht; er folgt nur seinem Kopfe. 16. Der Lehrer machte die Schüler darauf aufmerksam, wie wohl und gut Gott Alles in der Welt geordnet habe. 17. Der Richter fragte ihn vergebens, warum er dieses Verbrechen begangen habe; der Angeschuldigte hatte nichts darauf zu antworten. 18. Ich habe das Schreiben erhalten; allein ich weiß nicht, was ich darauf antworten soll. 19. Ich wüßte schon, was ich darauf antworten würde, wenn ich an Ihrer Stelle wäre. 20. Die Männer, von dezen Sie sprechen, sind eben nicht die besten Bertreter des Landes. 21. Ich ließ meiner Zunge freien Lauf, und erzählte. bas mir widerfahrene Unrecht. 22. Er ließ seiner Rede freien Lauf, und sagte in seiner Begeisterung mehr, als er hätte thun sollen. 23. Der An | Wunde schmerzt ihn mit jedem Lage mehr. 3. Ge schmerzt nichts mehr, Mir thut das schon weh, was andern nur leid thut. EXERCISE 144. 1. Es schmerzt mich, so viele Menschen unglücklich zu sehen. 2.- Die 經 als von Leuten verkannt zu sein, deren Liebe und Achtung man sich gern erwerben möchte. 4. Es thut mir leid, ihn beleidigt zu haben. 5. Schei, ben und Meiden thut weh, sagt ein altes deutsches Volkslied. 6. Der Kopf thut mir weh. 7. Gs thut mir in der Seele weh, ihm nicht helfen zu können. 8. Was fehlt Dir, mein Freund, warum so traurig? 9. Es fehlt mir weiter nichts, als daß ich ein wenig verstimmt bin. 10. Sind Sie frant? 11. Ja, ich bin ein wenig unwohl. 12. Was fehlt Ihnen? 13. Ich habe Kopfweh. 14. Sie sind reich und angesehen, und doch sind Sie niedergeschlagen, was fehlt Ihnen ? 15. Es fehlt mir viel, 3ufrieden. heit und Seelenruhe." 16. All (Sect. XL. 6) meine Freunde, die ver. sprochen hatten, zu kommen, waren da, nur Einer fehlte. 17. Alle Menschen fehlen. 18. Mein Bruder ist abermals fehl gegangen; statt in mein Haus, ift er in das meines Nachbars gekommen. 19. Seine Worte gereuten ihn, und er versprach, dieselben nie wieder sagen zu wollen. 20. Als dieses vor. fiel, war ich nicht zu Hause. 21. Dieser Streit fiel in der Nähe meiner Wohnung vor. 22. Ich habe dem bereits Gesagten nur noch wenig hinzu. zufügen. 23. Diesem Geschenke fügte sie ein kleines Briefchen hinzu. 24. Gr fügte mir vorsäßlich dieses Leid zu; teßhalb kann ich ihm nicht verzeihen. EXERCISE 145. 2. 1. It pains a father to hear of the wickedness of his son. Nothing pains more than to be accused innocently. 3. It pains me that so many persons have been found killed by the last storm. 4. I am sorry that you did not find me at home. 5. | The wound which the soldier received in the contest pains him. 6. What ails you, my friend? 7. Oh, nothing particularly. 8. You look very ill, what is the matter with you? 9. I am not well, I have hurt myself. 10. He has fallen out of the window. 11. This boy lacks understanding. 12. You have been offended by me; I am sorry, for I esteem you much. 13. You dare not lack courage to encounter the contest with your enemy. 14. I lack patience to await the result of this matter. SECTION LXXVI.--IDIOMS OF VARIOUS KINDS. Damit (therewith) is often to be rendered by "in order to, in order that, so that," etc., as ::—Ich muß eilen, damit ich nicht zu spät anfemme, I must hasten, in order not to arrive too late. 3ch wollte bitten, daß Sie das thäten, damit ich es nicht thun müßte (Gellert), I would beg you to do that, in order that I might not be obliged to do it. = 1. Seitdem since, since then, since the or that time; as:— Seitdem sie in Deutschland war, spricht sie nichts als Deutsch, since she was in Germany, she speaks nothing but German. Seitdem ist er glüdlich, since then (or that time) he is happy. 2. Gefälligst, an adverb in the superlative degree, from the adjective gefällig, pleasing, agreeable, answers to our phrase "please, if you please;" as :-Wollen Sie mir gefälligst* sagen, wieviel Uhr es ift? will you please to tell me what time it is? Geben Sie mir gefälligst meinen Hut, please to give me my hat. | Der Onkel feste seinen Hut auf und 2. The uncle put on his hat and cloak (hung his cloak round). He hastens home with anxious soul in order that he may not miss the appointed time. EXERCISE 146. 1. Seitdem ich hier angekommen bin, hat sich schon Manches ereignet, Seitdem er diese That begangen hat, scheint aller Friede von ihm ge. wichen zu sein. 3. Seitdem er fort ist, habe ich keine recht frohe Stunde mehr. 4. Seit dieser Zeit hat man nichts wieder von ihm gehört. 5. Seit meinem zehnten Jahre habe ich das elterliche Haus verlassen. 6. Seit gestern befinde ich mich nicht ganz wohl. 7. Seit dem Tode seiner Eltern irrt er ohne Heimath in der Fremde umber. 8. Seitdem er zur Erkenntniß seiner selbst gekommen ist, ist er ein ganz anderer Mensch geworden. 9. Er zog sich in aller Geschwindigkeit an. 10. In der Gile vergaß er seine Stiefel anzuziehen, und eilte in den Pantoffeln fort. 11. Seine Kleider waren ganz durchnäßt, deßhalb mußte er sich anders anziehen. 12. Er seßte diesen Morgen seinen Hut nicht auf, sondern seine Müße. 13. Der Diener hängte seinem Herrn nicht, wie gewöhnlich, den Mantel um, sondern er warf ihn sich selber um. 14. Bergessen sie nicht, Ihren Mantel umzuhängen, es ist sehr kalt und stürmisch. 15. Hängen Sie mir gefälligst meinen Mantel um, und seßen Sie mir meinen Hut auf, denn ich habe schon meine dicken Pelzhandschuhe angezogen. 16. Er stieg auf den höchsten Baum, damit er den König sehen könne. 17. Er war sehr eilig, damit er die Abfahrt des Postwagens nicht versäumen möchte. 18. Er erzählte mir dieses, damit ich mir ein Beispiel varan nehmen möchte. 19. Der Schüler entschuldigte sich damit, daß er keine Zeit gehabt hätte, seine Aufgabe zu lernen. 20. In großen Staaten müssen Hunderte hungern, damit Einer prasse und schwelge: Zehntausende werden gedrückt und in den Tod gejagt, damit ein gekrönter Thor over Weiser seine Phantasien ausführe. EXERCISE 147. 1. Will you please to give me a cup of coffee or tea? 2. Since yesterday I have felt myself not quite well. 3. Since he quitted his parental house we have not heard anything of him. 4. Since the twelfth year of my age I have not visited my native land. 5. Since he received the intelligence, he has had no peace. 6. In order that my friend may not come in vain, I shall stop at home. 7. I have not seen my friend since he arrived from Germany. 8. Instead of putting on his boots, he went out in his slippers. 9. Tell your friend, if you please, he may visit us at any time. 10. Why does he not take advantage of his youth, in order to acquire the knowledge he wants? 11. How have you been since I saw you last? 12. Finish your exercise, if you have not yet finished it, then you will not be punished by your master. SECTION LXXVII.-IDIOMS OF VARIOUS KINDS (continued). Fieb, beloved, dear, agreeable, may, when applied to persons, be rendered (like gern with haben, Sect. XLIII. 1) "dear;" as:Seittem'. (See R. 1, Ich habe ihn sehr lieb, he is very "dear" to me. Applied to things, lieb with sein signifies to be agreeable, to please, etc.; as:-Dieses kleine Geschenk ist mir lieb, this little present pleases (is pleasing) Es ist mir lieb, daß Sie damit zufrieden sind, me, or is dear to me. above.) Um'werfen, to throw round, put on. Versau'men, to miss, neglect, lose. Vortheil, m. advan tage. Pelz'handschuh, m. fur- carouse. fore. The Germans can only (for the He is clothed with the garment • Note, that gefälligst, unlike the phrase Sect. LXXIV, 1, is employed In soliciting and not in acknowledging a favour. I am glad (it is pleasing) that you are satisfied with it. 2. Kennen lernen (kennen, Sect. XXV. 3) signifies "to become ac quainted with." Wollen Sie ihn kennen lernen? do you wish to become acquainted with him? Ich habe ihn schon kennen gelernt, I have already become acquainted with him. EXERCISE 148. 1. Es ist mir lieb, daß ich Sie hier antreffe; ich habe Ihnen Wichtiges mitzutheilen. 2. Es ist mir lieb, Sie so wohl zu sehen. 3. Es wäre mir lieb, Sie bald wieder zu sehen. 4. Er ist böse über das Betragen seines Neffen. 5. Er ist böse über das Ausbleiben seines Sohnes. 6. Sie ist böse über sich selbst. 7. Der Freund war böse auf mich, aber ich habe ihn wieder besänftigt. 8. Die Mutter ist böse auf ihr eigensinniges Kind. 9. Ich bin böse auf ihn, weil er mich beleidigt hat. 10. Kennen Sie Herrn N.? 11. Ja, ich habe ihn lezte Woche in dem Hause Ihrer Frau Lante kennen gelernt. 12. Ich lerne ihn mit jedem Tage mehr kennen. 13. Man lernt Jedermann eher kennen, als sich selbst. 14. Wo sind Sie mit diesem Herrn bekannt geworden? 15. Wir fennen uns von Jugend auf, and lernen uns mit jedem Tage mehr kennen. 16. Kennen Sie Frau lein B.? 17. Nein, aber ich hoffe noch mit ihr bekannt zu werden. 18. Dieser Mann wird durch seine trefflichen Werke bald bekannt werden. 19. Herr N. stellte mich dieser Familie vor. 20. Er wurde der Gesellschaft durch seinen Bruder vorgestellt. 21. Das Acußere dieses Mannes ftellt nichts vor. 22. Dieser Schauspieler stellte Karl XII. vor. 23. Durch wen wurten Sie eingeführt? 24. Ich verdanke (Sect. XLIII. 6) diese Ehre der Nichte des reichen Kaufmannes. 25. Der Freund führte mich in die Gesellschaft ein. 26. Der Franzose führte diese neue Mode ein. 27. Der Schmuggler führt verbotene Waaren ein. EXERCISE 149. orderly life. 9. But I remind you of his actions in the last war, of which he may justly boast. 10. Rejoice at his acquittal, and vouchsafe him your friendship. 11. Do not scoff at him because he was in the dungeon, but rather pity him and think of his sufferings. 12. Let every one who laughs at him be ashamed of his own behaviour." 13. All present rejoiced at this speech, and they instantaneously released the accused man of his fetters. 14. Teach us thy way, O Lord! and deprive us not of thy grace. 15. I intend to visit a watering-place next summer. 16. I cannot by any means be of opinion that one should not indulge in repose after dinner. 17. He who rejoices at life should also remember death. 18. When General Tilly had conquered the town of Magdeburg, he laughed at the supplicants who besought him to commiserate them. 2. EXERCISE 97 (Vol. II., page 95). 6. Go ge 1. In früheren Zeiten konnten die Leute nicht lesen, viel weniger schreiben Ich bin Willens, im nächsten Sommer die Bäter Homburg und Laubach zu besuchen. 3. Als Ludwig Philipp, König ter Franzosen, seinem Throne entsagt hatte, ging er mit seiner ganzen Familie nach England. 4. Ginige Könige haben wenig Ursache, sich ihrer Regierung zu rühmen. 5. Kaiser Karl V. entsagte seiner Krone und ging in ein Kloster. ziemt einem Manne besser, auf seine Handlungen aufmerksam zu sein, als sich seiner Fähigkeiten zu rühmen. 7. Ich werde mich aller meiner Geschäfte entledigen, und ein ruhiges Leben genießen. 8. Er freute sich ter Freisprechung der Unschuldigen, und würdigte sie der größten Freundschaft. 9. Die Feinde stürmten die Stadt, und lachten der Flehenden, welche sie baten, sich ihrer zu erbarmen. EXERCISE 98 (Vol. II., page 118). 1. The old Saxons abjured their gods after Charles the Great had completely vanquished them. 2. Whilst he foreswore this deed with a false oath, he denied the immortality of the soul. 3. His wickedness is perceptible in his eyes. 4. I have not given up the hope of again seeing my relations. 5. I had ordered my servant to call me as soon as you came. 6. When I heard myself called, 1 turned back immediately. 7. If I were to imitate you, I should soon have no more money. 8. It has often happened to him already, that he looked for his spectacles and had them upon his nose. 9. The king passes through this town to-day. 10. Steer thy ship through the raging waves, courageous pilot. 11. A judicious father checks the rude behaviour of his children in time. 12. I seldom have money, but always debts; I wish I only knew how to check this inconvenience. 13. For what does it profit a man if he gain the whole world and lose his own soul? 14. His strength assisted (him) to endure these suffer15. The cook tasted of the dishes. 5. My ings. 16. It cost me liberty and fatherland. 17. It cost him his first-born son. 18. He assured me of the truth of this circumstance. 19. They secured the thief. 20. One should seek to imitate good manners. 1. It would be very agreeable to me if you could leave me to myself. 2. It was very satisfactory to me to see my brother well. 3. I am very glad to hear that your undertaking has succeeded. 4. He is angry at the conduct of his brother. brother introduced me to Mr. G. 6. Has your sister already become acquainted with my brother? 7. Yes, she became acquainted with him at the last concert. 8. Do you know why your brother is so angry? 9. He is angry at me, because I laughed at him. 10. The actor personated Henry IV. very well. 11. That government has introduced good laws. 12. This fashion has been introduced by the French. 13. The import of wine from France is very great. KEY TO EXERCISES IN LESSONS IN GERMAN. 1. Ich schmeichle mir, daß Sie mich mit einem Besuch beehren werden. 2. Er würde mir gewiß schaden, wenn er mir beikommen könnte. 3. Der Stern der Hoffnung leuchtet dem Menschen selbst in der düstersten Nacht. 4. Ich werde meinem Freunde beistehen für die Hülfe, welche er mir geleistet hat. 5. Der Lehrer hat mir gesagt, er sei mit seinen Schülern zufrieden; sie seien fleißig, und kämen allen seinen Wünschen zuvor. 6. Ich kenne keinen, welcher den Leidenschaften Anderer so schmeichelt, als er. 7. G8 wäre zu wünschen, daß Jedermann den Armen beistehen möchte. 8. Schmeichle veinen Kindern nicht zu viel. 9. Navoleon sammelte die besten feiner Generäle um sich. 10. Nachrem er denselben sein Vorhaben mit. getheilt hatte, erboten sie sich ihm beizustehen. 11. Er traute seiner eigenen Macht, überzog Europa mit feindlichen Truppen, und troßte jeter Gefahr. 12. Seiner eigenen Meinung nach war seine Macht unumschränkt, und er gedachte nicht der Schwierigkeiten, welche ihn umgaben. EXERCISE 96 (Vol. II., page 95). MECHANICS.-III. FORCES APPLIED TO A SINGLE POINT-PARALLELOGRAM FROM the principles stated in our second lesson, it is evident Take, for instance, three equal weights, attached to three strings, two of them much longer than the third, which are tied together in a knot at their other ends. If the two longer strings with their attached weights are now thrown over two pulleys in the same plane, one of the pulleys being even higher up than the other, and the third string and weight is allowed to hang down in the middle, we shall have a case of three equal forces applied to a point. There are the two outside weights acting over the pulley, and drawing the knot obliquely to either side, 1. I suppose you still remember the young man who was accused and the middle weight pulling it downwards. What position of robbery last year. 2. He was accused of having robbed a rich will the strings settle themselves into? Evidently so that the cattle-dealer of his money on the highway. 3. But they could not angles between the strings may be equal; for no reason in the convict him of this crime. 4. He had already given up all hope of an world can be given why they should be unequal. Whatever reason acquittal, and abandoned the idea of being declared innocent. 5. could be assigned for supposing one of these angles greater than The judge, however, relieved him of all anxiety. 6. After he had the other, that same reason should make that other angle told the accused man to be of good heart and cast away all sorrow, he said, "I am fully of opinion that they cannot charge this young man greater than the first. The angles, therefore, must be equal. with the robbery. 7. For it is not every one who is ashamed of beg thence of any number of forces, applied to a single point may We have next to consider how the resultant of two, and ging, and destitute of all means, that becomes a robber. 8. I can speak highly of his behaviour, for he has always been addicted to an be found. You will keep in mind that by a “single point" I D с mean a point "in a body;" and that will save me always adding the latter words when I use the former. Of course, forces applied to "a material point" are included in the description, and these you will find to be of great importance. As the joint effect of two or more forces so applied is termed their "resultant," so we name the separate forces of which it is the effect its components. There are thus two operations, the Composition of Forces, and the Resolution of Forces, with which we may be concerned: by the former we denote the putting together, compounding, or finding the resultant of any number of forces, and by the latter the separating, or resolving, of any given force into the two or more to which it may be considered equivalent. The Composition we first consider; but this requires a short digression on the nature of a parallelogram. A parallelogram is a four-sided figure whose pairs of opposite sides and opposite angles are equal. In the adjoining figure, ABCD is a parallelogram, if the side A B is equal to D c, and also B C to A D. The two cross lines, A C and B D, are called the "diagonals of the parallelogram." Now, if you examine the two triangles, A B C, A DC, which are on opposite sides of the diagonal, A c, you will see reason for believing that they must be equal to each other. They are, in fact, the same triangle on opposite sides of that line; for they have A C for a common side, and the two other pairs of sides are equal, namely, A B equal to D C, and A D to BC; and you cannot out of three straight lines make two different triangles. Now, the point to which I am trying to lead you, and which you will soon find of importance, is that, since these triangles are equal-in fact, one and the same triangle in two positionstheir angles must be equal to each other. Hence we arrive at the following important properties of a parallelogram: Fig. 1. B 1. That the opposite angles, A B C and A D C, are equal, also the opposite angles, B A D and B C D. 2. That either diagonal makes equal angles with the pairs of opposite sides, A B D equal to C D B, and A D B equal to C B D. It is on account of this latter property the figure is called "parallelogram." The opposite sides are not only equal, but parallel, on account of their making equal angles with either diagonal. However, keep in mind that these angles are equal, for this knowledge is necessary to your properly understanding what we next come to, namely THE PARALLELOGRAM OF FORCES. The forces in our cuts and diagrams being represented, as agreed on, by lines, and their directions by arrow-heads attached to their remote ends, this principle may be stated as follows:If two forces applied to a point are represented in magnitude and direction by two straight lines, their resultant is represented in magnitude and direction by the diagonal passing through that point of the parallelogram of which these lines are two adjoining sides. P R In Fig. 2 let o P, o Q be the two forces, and draw from P and Q the two dotted lines parallel to them which meet in R, then the dotted diagonal, o R, of the parallelogram thus formed is the resultant, both in magnitude and direction, of o r and o q. Now, I shall not here give you the strict mathematical proof of this -proposition; it is too complicated, and involves so much close reasoning, that to force it on a student in the beginning of a treatise on mechanics would be to throw an unnecessary difficulty in his way. The best course is to defer it until you have become more accustomed to mechanical reasoning, and then return to it. In the meantime you can satisfy yourselves that it is true by a reference to the two following experiments, one derived from equilibrium, the other from motion. Fig. 2. First Experiment.-Let three weights, U vw, be attached to three cords, as in Fig. 3, which are knotted together at o; and let two of the cords, longer than the third, with their at R tached weights, be thrown over two pulleys, P Q, which move freely in the same plane round axles fastened into a wall or upright board. Arrange, then, the weights and cords until equilibrium is produced. It is evident, from the principle stated at the close of the last lesson, that the force, w, must be equal and opposite to the resultant of u and v, acting over the pulleys at o. Now, take on P the cord o P, a length o A, equal in inches to the number of pounds in U, and on o Q another, o B, equal to the pounds in v, and then draw the parallels, A R and B R, to o P and o Q, meeting in R; OR will then be the resultant of u and v, if the principle of the parallelogram of forces be true. It should, therefore, be opposite in direction to the force w, and the number of inches in it should be equal to the number of pounds in w. Now, on trial it is found that o R is opposite to w, that is to say, that it points vertically upwards in the plomb-line; and it is also found that the number of inches in its length is that of the pounds in w. W Fig. 3. Second Experiment.-Let us suppose that a parallelogram O A E B is described anyhow on a perfectly smooth horizontal table, and that at the point o, two springs are fitted so that one of them, on being let go, would make the unit ivory ball move over o A in the same time that the other would make it move over OB. It is evident that the lines o A and O B would then represent these forces. Furthermore, it should follow, if the principle of the parallelogram of forces be true, that, when both springs are let go together so as together to strike the ball, it should move over the dia gonal o R of the parallelogram in the same time as the ball moved over o A and OB wherstruck separately. Now, this is what, on trial, exactly hap pens. The ball does move over the diagonal, and moves over it in the same time that it previously moved over the sides. This it could not do if the resultant of two forces was not represented in magritude and direction by the diagonal. Instruments are fitted up for lecture-rooms by which the experiment can be made, and the result always is as I have stated. A R Taking the principle, then, as established, let us observe its consequences. You are given two forces, acting at a point, and you want their resultant. Make, you will immediately say, a parallelogram of the two forces, and the diagonal is the required line. Not so fast; you need not describe the whole of that figure, a part will suffice. Now, if from the end A of o A, you draw A R parallel and equal to o B, it is clear you do not want to draw BR at all. A R gives you the far end of the resultant, and all you have to do then is to join R with o, and your object is o gained. Thus your paralFig. 5. lelogram of forces suddenly becomes a triangle of forces; and you may lay this down as your rule in future for compounding two forces. B |