of "Finden's Tableaux :" and a series of papers called "Readings of Poetry, Old and New." The cones corresponding to each stripe have the same axis, which is called the axis of vision. This straight line being parallel to the rays of the sun, it follows that when the sun is in the horizon, the axis of vision is itself horizontal, and the rainbow appears in the form of a semi-circle. If the sun is raised above the horizon, the axis of vision is lowered, and with it the rainbow. Lastly, when the sun is 42° 2′ above the horizon, the rainbow disappears altogether below the horizon. Hence it is never seen except in the morning or evening. All that we have stated applies to the inner bow. With regard to the outer one, it is formed by rays which have undergone two reflections, as is seen in the ray s' idfeo in the drop p. The angle Io, formed by the emerging and incident rays, is called the angle of deviation, as before. In the present case, however, this angle is not susceptible of a maximum, but of a minimum, which varies according to the different kinds of rays, and which has effective rays corresponding to it. It is proved by calculation, that for violet rays the minimum angle is 54° 7′, and for red rays only 50° 57', which explains why the red bow is in this case inside and the violet outside. As at each interior reflection in the drop of water there is a loss of light, the outer bow always exhibits fainter colours than the inner one. The outer bow ceases to be visible when the sun is more than 54° above the horizon. The moon produces rainbows as well as the sun, which are historical character than her springing spaniel, or Italian greycalled lunar rainbows, but they are very pale. BIOGRAPHY.-No. XXVIII. MARY RUSSELL MITFORD. THIS lady was born on the 16th of December, 1786, at Abres- Although her tragedies show great intellectual powers, and a highly cultivated mind, yet it is by her sketches of English life that she has obtained the greatest share of her popularity, and it is on them her fame will chiefly depend. In these descriptions Mary Mitford is unrivalled. Her manner is inimitable and indescribable, and sheds interest around the most homely subjects and coarsest characters. Who ever threw by a sketch of hers half read? No one who admired a spring daisy-or that most fragrant blossom, the wallflower, which beautifies every object, however rough, rude or ruinous, around which it wreathes. And though she does not trace the motives of conduct very deeply, or attempt to teach principles of moral duty, yet there is much in her sprightly and warm sketches of simple nature which draws the heart to love the Author of all this beauty; and much in her kind and contented philosophy to promote love and good feelings. She was a philanthropist, for she took pleasure in the happiness of others a patriot, for she drew the people to feel the beauties and blessings which surround the most lowly lot in the "land of proud names and high heroic deed." "As a proof that we love her, we love her dog," says an American writer. "Walter Scott's stately Maida is not more an hound. If she began by being prosaic in poetry, she has redeemed herself by being most poetic in pastoral prose." In 1833 Miss Mitford's name was added to the pension list, a well-earned tribute to one whose genius has been devoted to the honour and embellishment of her country. After a long period of decline and helpless suffering, cheerfully borne, this eminent lady died lately at Swallowfield Cottage, near Reading, aged six v-six years. WHITSUN-EVE-MY GARDEN. "The pride of my heart and the delight of my eyes is my. garden. Our house, which is in dimensions very much like a bird-cage, and might, with almost equal convenience, be laid on a shelf, or hung up in a tree, would be utterly unbearable in warm weather, were it not that we have a retreat out of doors, and a very pleasant retreat it is. To make my readers fully comprehend it, I must describe our whole territories. cottage at one end; a large granary, divided from the dwelling Sphynx ligustri, privet hawk-moth. Hence, Cos. CD: R:: cos. B: sin. B C D. Cor. 3. The sines of the segments of the base are reciprocally proportional to the tangents of the angles at the base. For, by Theorem II., Also, sin. ADR:: tan. CD: tan. A. sin. BDR :: tan. CD: tan. B. sin. A D sin. BD:: tan. B: tan. A. Cor. 4. The cotangents of the two sides are proportional to the. cosines of the segments of the vertical angle. For, by Theorem II., Cor. 2, hottest days over the sweetest flowers, inserting its long pro-Alsɔ, "What a contrast from the quiet garden the lively street! Saturday night is always a time of stir and bustle in our Village, and this is Whitsun-Eve, the pleasantest Saturday of all the year, when London journeymen and servant lads and lasses snatch a short holiday to visit their families. A short and precious holiday, the happiest and liveliest of any; for even the gambols and merry-makings of Christmas offer but a poor enjoyment, compared with the rural diversions, the Mayings, revels, and cricket-matches of Whitsuntide.' LESSONS IN TRIGONOMETRY.-No. V. OBLIQUE-ANGLED SPHERICAL TRIANGLES. THEOREM III. In any spherical triangle, the sines of the sides are proportional to the sines of the opposite angles. In the case of right-angled spherical triangles, this proposition has already been demonstrated. Let, then, A B C be an oblique-angled triangle; we are to prove that sin. B C : sin. A c :: sin. A: sin. B. Through the point c draw an arc of a great circle CD perpendicular to A B. Then, in the A B cos. A CD cot. A c:: tan. CD: R. Also, COS. A CD cos. BCD cot. A c cot. B C. Let B Ca, Ac=b, BD=m, and AD=n. Then, by Theorem III., Cor. 1, cos. a: cos. b:: cos. m : cos. R. Whence, COS. C s.a+cos. b: cos. a—cos. b:: cos.m+cos. n: cos. m—cos. n. But by Trigonometry, cos. a+cos. b: cos. a— cos. b:: cot. 1(a + b): tan. 1(a - b). Also, cos.m+cos. n: cos.m—cos. n:: cot. 1(m+n): tan. 1(m—n). Therefore, cot. (a+b): cot. 1(m + n) :: tan. (a - b) : tan. 1(m — n) But, since tangents are reciprocally as their cotangents, we have, cot. (a+b): cot. 1(m + n) : : tan. 1(m + n): tan. 1(a + b). Hence, tan. (mn): tan. (a + b) :: tan. (a - b) : tan. †(m — n)、 In the solution of oblique-angled spherical triangles, six spherical triangle AC D, right-angled at D, we have, by Napier's cases may occur, viz.: rule, 1. Given two sides and an angle opposite one of them. 2. Given two angles and a side opposite one of them. 3. Given two sides and the included angle. 4. Given two angles and the included side. 5. Given the three sides. 6. Given the three angles. sin. A sin. A CB:: sin, BC: sin. A B = 145° 5' 0". When we have given two sides and an opposite angle, there are, in general, two solutions, each of which will satisfy the conditions of the problem. If the side a c, the angle A, and the side opposite this angle are given, then, with the latter for radius, describe an arc cutting the arc A B in the points в and B. The arcs CB, CB' will be equal, and each of the triangles ACD, ACB' will satisfy the conditions of B' D B triangles A B C, A B' O will satisfy the conditions of the problem. There is the same ambiguity in the numerical computation, since the side BC is found by means of its sine. In the preceding example, however, there is no ambiguity, because the angle A is less than B, and therefore the side a must be less than 6, that is, less than a quadrant. Ex. 2. In the oblique-angled spherical triangle ABC, the angle A is 128° 45', the angle c=30° 35′, and BC= 68° 50'. Required the remaining parts. AB Ans. AC B 37° 28′ 20′′. 40° 9' 4". 32° 37′ 58′′. CASE III. Given two sides and the included angle, to find the remaining parts. In the triangle A B C let there be given two sides, as A B, A C, and the included angle A. Let fall the perpendicular o D on the side AB; then, by Napier's rule, R COS. A tan. AD cot. A C. then, by Theorem III., Cor. 3, Having found the segment' A D, the segment BD becomes the problem. There is the same ambiguity in the numerical Ex. 2. In the spherical triangle ▲ B C, the side a = 124° 53′, b=31° 19′, and the angle ▲ = 16° 26'. Required the remaining parts. A C D sin. B: sin. A:: Ex. 1. in the oblique-angled spherical triangle ABC, there are given the angle = 52° 20', в 63° 40', and the side 683° 25'. Required the remaining parts. sin. B sin, A:: sin. A C· sin. B C 61° 19′ 53′′. Then, in the triangle ▲ C D, cot. ACR cos. A tan. A D 79° 18′ 17". The remaining parts may now be found by Theorem III. Hence, cot. A c: cos. A::R: tan. A D 37° 33′ 41′′. sin. BD sin. A D :: tan. ▲ : tan. B 31° 33′ 43′′. Also, by Theorem III., Cor. 1, cos. A D cos. BD:: cos. AC: Cos. BC= 40° 12′ 59′′. Then, by Theorem III., sin. Bo sin. AB:: sin. A: sin. ACB = 131° 8′ 46′′. Ex. 2. In the spherical triangle ABC, the side A B 78° 15', AC 56° 20′, and the angle ▲ = 120o. Required the other parts. CASE IV. Given two angles and the included side, to find the remaining parts. In the triangle ABC let there be given two angles, as a and ACB, and the side A c included between them. From o let fall the perpendicular CD on the side A B. Then, by Napier's rule, R COS. AC cot. A cot. A CD. Having found the angle ACD, the angle BCD becomes known; then, by Theorem III., Cor. 4, COS. A CD: COS. BCD:: cot. AC: cot. BC. The remaining parts may now be found by Theorem III. Ex. 1. In the spherical triangle ABC, the angle = 32° 10′, the angle A CB 133° 20′, and the side A039° 15'. Required the other parts. Also, by Theorem III., Cor, 2, sin, ACD: sin. BCD:: cos, A : cos. B 28° 15′ 47′′. Then, by Theorem III., sin. B: sin. ACB:: sin. AC: sin. A B 76° 23′ 5′′. Ex. 2. In the spherical triangle A B C, the angle a=125° 20′, the angle o 48° 30', and the side AC 83° 13. Required the remaining parts. AB Ans. BC 56° 39' 9". 114° 30′ 24′′. B = 62° 54' 38". The specimen of an interrogative sentence given above, is that of a direct interrogative. The sentence is called a direct interrogative sentence, because it simply and directly asks a question. If, however, I put the sentence thus, “I do not know whether the man is good," then I form what is called an indirect interrogative sentence; that is, a sentence in which not know whether the man is good," is equivalent to, "Is the a question is implied or involved. Thus the sentence, "I do man good?" and, "I do not know." An indirect interrogative sentence is, consequently, a compound sentence. Interrogative sentences are formed in Greek by means of interrogative words. Such words are numerous in Greek, if CASE V. Given the three sides of a spherical triangle, to find the only because the language has two forms of words, one for the angles. direct and one for the indirect interrogatory. The indirect interrogatives are formed from the direct, by prefixing to the latter the syllable o, by which it is indicated that the question rests on the foregoing sentence or clause. In the triangle ABC let there be given the three sides. R CO3. A tan, A D cot. A C. The other angles may now be easily found. A sentence is formed by the junction of a predicate with a and you ask of what subject a certain thing is to be predicated. Hence arise what are called noun-questions; that is, questions in which you ask for the noun. But your question may relate also to the predicate, and you then ask what is said Ex. 1. In the spherical triangle ABC, the side AB=1129 25', of the subject? Hence, who and what involve the substance Ac 60° 20', and BC= 81o 10'. Required the angles. From which, by Case V., the angles are found to be, LESSONS IN a 98° 21′ 20′′. Ans. b 109° 50' 10". c = 115° 13' 7". GREEK.-No. LXIL By JOHN R. BEARD, D.D. INTERROGATIVE AND IMPERATIVE SENTENCES. INTERROGATIVE sentences are simple sentences put in the way of a question. Thus, instead of saying, "The man is good," I say interrogatively, "Is the man good?" But this interrogation, though in itself a simple sentence, is incomplete, inasmuch as for its completeness it wants an answer. It is thus seen that interrogative sentences form the transition point between simple and compound sentences. Causal interrogatives, or such as ask for the cause or reason of a thing, are formed by the interrogative pronouns, in connection with a preposition; as, dia ri; why? Toù éveka; on what account? ET; on what condition? Indirect, ctori, ότου ένεκα, αφ' ότῳ, etc. In the same way are formed some of the temporal interrogatives; as, μexpι τov; how long? so, μέχρι ποσου ; μέχρις όσου. Sometimes the direct interrogative is employed instead of the indirect, the question being put independently: e.g είπε μοι, ποῖον τι νομίζεις ευσέβειαν ειναι ; tell me, what do you consider piety to be? That is, "What do you consider piety to be? pray tell me." Sometimes the direct and the indirect are connected together: e.g. ου γαρ αισθανομαι σου, ὁποιον νομιμον η ποιον δικαιον λεγεις I do not learn from you what you call lawful, or what just. In the interchanges of conversation, the indirect question answers the direct in asking the question indirectly: eg. Interrogation requiring a Negative Answer. άρα μη ιατρος βουλει γενέσθαι surely you do not wish to become a physician? "No." When a double question, or a question with an alternative member, is asked, άρα is superseded by ποτερον οι ποτερα, which merely intimates that the question relates to two mutually excluding points: e.g. πότερον μονῳ μοι βουλει διαλεχθηναι, η και μετα των αλλων; | do you wish to speak to me by myself, or together with the rest? 2., truly, shows that the interrogator has a special interest in ascertaining the true state of things, and may, consequently, be often rendered by "in truth," "really; but it is often untranslateable: e.g. ἤ οὗτοι πολεμοι εισιν ; are those enemies? Other particles are adjoined to . These strengthen the original one, as, ǹ dη, ǹ dñra; or they weaken its force, as, TOU; or they ground the question on something else, and so give it more emphasis, as, yap, for truly; which, when it stands as an independent elliptical sentence, may be rendered, is it not so ? 3. pwv (formed out of un ovv, is it not then?), indicates that the inquirer supports his question on something (ovv, then, therefore), but is uncertain whether thereby he shall hit the truth. Consequently, this particle conveys some uncertainty, and may be rendered by "perhaps," "it may be": e.g. μῶν τί σε αδικει ; perhaps something injures thee? So with the negatives, puv ov, "does not?" "Yes." μwv un, "surely not?" "No." 4. alλo rin, forms the direct opposite of μ, literally, is it other than? which amounts to a strong affirmation; the form intimates that the questioner has hit the right view of the matter, and accordingly expects the opposite (also indicated by allo ri) to be unconditionally denied: e.g. αλλο τι η αδικοῦμεν ; beyond a doubt, we are acting unjustly. Then is sometimes dropped without a marked alteration in the sense: e.g. αλλο τι ; γεωργος μεν εἷς, αλλος δε τις ύφαντης 5. εἶτα and επειτα, also κᾆτα (και ειτα), and καπειτα, δε έλθη, represent the question as called forth by something which excites surprise or dissatisfaction in the interrogator, and may often be Englished by, and now? and yet? what! e.g. επειτ' ουκ οιει θεους ανθρωπων φροντίζειν ; and so then, you do not think the gods have care of men? 6. Ti paw (literally, learning what?), ri alov (literally, suffering what?), express dissatisfaction in the fact implied in the question: e.g. τι μαθών καταφρονεῖς των αμεινόνων ; In regard to moods, the same rules obtain for interrogative sentences as for affirmative. Consequently the indicative is employed when the questioner inquires after a fact as the basis of an opinion or judgment. If inquiry is made in regard to a circumstance which the inquirer does not regard as existing, since, in his view, the condition necessary to its existence does not exist, then, in the interrogative sentence, the indicative of the historical tenses with av, is employed. A sentence thus constructed has its basis in a supposed preliminary sentence, which is commonly expressed, but sometimes is merely understood, or implied from the tenor of what is said: e.g. ει τις σε ήρετο, όπως εσώθης, τί αν απεκρίνω ; if any asked thee how thou wast saved, what would you answer? The subjunctive appears in direct questions when a person directs a question to himself, so as to give the idea that he is undetermined as to what opinion he should form of the matter; as, ειπωμεν η σιγῶμεν ; shall we speak, or be silent? The subjunctive takes av with it when reference is made to a completely hypothetical foregoing proposition : πῶς αν εν φρονησαντες ταῦτα καλῶς ἔχειν ἡγησωνται ; τί ποτ' αν οὖν λεγωμεν ; what, then, should we say The optative, in direct questions, denotes the same kind of uncertainty as the subjunctive, only in regard to circumstances which appear as belonging to the past, whereby the matter looks very doubtful for the present: e.g. τι αίσχιον και κακιον ειη ; what could be more shameful and base? The particle av, as in the subjunctive, increases the uncertainty: e.g. τίς ουκ αν μαινεσθαι ὑμας νομίσειεν ; who would not consider you mad? In indirect interrogative sentences the employment of the moods is generally governed by the same rules as in direct interrogative sentences. The province of the indirect interrogative is more extended in Greek than in English, for verbs expressive of fear or care take after them a sentence (or clause) of that kind. The mood depends on the tense of the principal verb, and on the degree of doubt or uncertainty; even the indicative may be used after the accompanying μn, if full conviction is intended: e.g. νῦν φοβούμεθα μη αμφοτέρων ἅμα ἡμαρτηκαμεν now we fear, lest we have at once missed both. As peculiarities in the construction of interrogative sentences, observe a kind of intermingling of the indirect sentence and the principal sentence: a.g. το των χρημάτων, ποσα και ποθεν εσται, μαλιστα ποθεῖτε |