1. Let AB be the given right line. On BA produced describe a triangle AEC, equal in all respects to the given triangle (I. 22). Through E draw GK parallel to AB. Bisect AC in D and draw DG perpendicular to AC, and meeting GK in G. Through A draw AH parallel to DG. Complete the parallelogram HABK. Join KA, and produce KA till it meets GD produced in L. Through L, draw LN parallel to AB, meeting HA produced in M, and KB produced in N. Then ABNM is a rectangle equal to the given triangle. By construction, the straight lines GL, HM, and KN are parallel, and also the straight lines GK, DB, and LN; and KL is a diameter of the parallelogram KNLG; consequently the complements AN and AG of the parallelograms BH and MD are equal (I. 43). By construction, the parallelogram, AG is equal to the triangle AEC (I. 42). Hence the parallelogram AN is equal to the triangle AEC, and consequently equal to the given triangle. AN is also a rectangle since the angle BAM is equal to the angle ADG, which by construction is a right angle. 4. Let AB be the given straight line. On AB describe the semicircle ADB, having its centre at C. Through C draw CD perpendicular to AB, and in it take a point E such that CEtwo-thirds of CD. Through E draw LK parallel to CB meeting the semicircle in K. From K draw KL perpendicular to AB, then the point L divides the straight line AB so that the rectangle under AL and LB is equal to the square on one-third of AB. Join CK. Then because the straight line AB is divided into two equal parts at the point C, and into two unequal parts at the point L. AL.LB+CL'=CB' (II. 5). But CB=CK, therefore AL.LB=CKBut CE-two-thirds of CL'=LK'=CE'. CD or one-third of AB. Thus the straight line AB has been divided in L, so that the rectangle AL.LB is equal to the square on one-third of AB. Q. E. F. Thus a rectangle AN has been described on the right line AB which is equal to the given right line. 2. This is Euclid II. 11. 3. We have (I. 47) AE-AD-DE' AC' +CD2-DE-CB2+CD2-DE-5 CD-DE2 =5DB2-DE2. But DE BE, therefore DB2=2 DE'. Consequently AE-9 DE, and therefore AE=3 DE. Q. E. D. 5. Let A be the given point, BC the given right line, and H the given radius. Draw the line DE parallel to AB at a distance from it equal to H. From A as centre at a distance equal to H describe the cirle KLM, cutting DE in L. From L as centre with the radius LA, describe the circle ANPQ. Then the circle ANPQ passes through the point A, touches BC, LP perpendicular to BC. By construction LP and LA are each equal to H. Therefore LP is a radius of the circle NAPQ, and since BC is perpendicular LP, BC touches the circle ANPQ in the point P (III. 16). Thus the circle NAPQ has been drawn through the point A, touching BC in P and of the radius LA equal to the given radius H. Q. E. F. 6. Let AB be the given base of the triangles. Upon AB as base erect an isosceles triangle ABC having the angles AB at its base each equal to the complement of the given vertical angle.* From C as centre with the radius CA, or CB describe a circle. The circle so described is the required locus. For, since the sum of the angles of the triangle ACB is equal to two right angles, the angle ACB must be equal to twice the given angle. But, if Eis any part on the circles described, the angle ACB is twice the angle AEB (III. 20). Hence the angle AEB of the triangle AEB is equal to the given vertical angle, and consequently the circle ABE is the locus of the verices of all triangles on the base AB and having the given vertical angle. The theorem on which this depends is the twentieth proposition of the third book of Euclid. 7. Inscribe a hexagon in the given circle (IV. 15.) Let AB be one of the sides of the hexagon. Bisect the arc AC in D. Bisect the arc AD in E. Then the arc AE is one-eighth of the arc AB and consequently one-fifty-eighth of the whole circumference of the circle. Therefore if the straight line AE be drawn and straight lines equal to it be placed round in the whole circumference of the circle (IV. 1), a regular polygon of forty-eight sides will be inscribed in the given circle. 8. A demonstration of this proposition is given on page 212 of the COMPETITOR. 9. This is Euclid, II. 36. 10. Let the diagonals of the parallelogram intersect at E. In the triangles DEA and BEC, the side AD is equal to the side BC, and the angles EDA and EAD are equal to the angles EBC and ECB respectively (I. 29); therefore the side ED is equal to the side EB (I. 26). Because the triangles AED and AEB stand on equal bases DE and EB, and are of the same height, the triangle AED is equal in area to the triangle AEB (I. 38). The complement of an angle is the difference between the angle and a right angle, For the same reason the triangle DOE is equal in area to the triangle BOE. Hence the sum (or difference) of the triangles AED and OED is equal to the sum (or difference) of the triangles AEB and BOE. Consequently the triangle ADO is equal to the triangle ABO. Q. E. D. ENGINEER STUDENTS. COPYING. Copy as much of the following paragraph as you can in ten minutes, taking care to copy neatly and cor. rectly, and to write as well as you can. N.B.-Good hand-writing is held to consist chiefly in the clear formation of the letters of the alphabet. MACCLESFIELD SILK TRADE, 1882. The demand for goods in the following department of the Macclesfield silk trade has been very active during the past year. In hand-made goods buyers have given their attention principally to the best descriptions of fabrics, which are just now being turned out by an unusually large amount of labour. In other descriptions of goods the operations have been characterised by more than the usual briskness, and the labour employed is very large. As regards the home trade, the purchases from stock have been numerous, and the enquiry for all classes of goods shows that some of the principal manufacturers will, for some time at all events, be very well employed. More favourable reports are received as to the trade at Congleton, and at Leek there are more enquiries than have been for some time past. Returns have come to hand showing a considerable increase in the amount of raw silk imported from Japan. This fact points to an expected increase in the demand for light summer silks. If the season is fine these provisions may be realized. DICTATION (1). The dreary scene of desolation, through which the traveller's road had lain on the preceding evening, was excluded from the view by some rising ground; and the landscape showed a pleasing al ternation of hill and dale, intersected by a river, which was in some places visible, and hidden in others, when it rolled between deeper wooded banks. The spire of a church, and the appearance of some houses, indicated the situation of a village at the place where the stream had its junction with the ocean. The vale seemed well cultivated, the little enclosures into which they were divided skirting the bottom of the hill, and sometimes carryiny their lines of straggling hedge-rows a little way up the ascent. Above these were green pastures, tenanted chiefly by herds of black cattle, then the staple commodity of the country, whose distant lowing gave no unpleasing animation to the landscape. The remoter hills were of a sterner character; and, at a still greater distance, they swelled into mountains of dark heath, bordering the horizon with a screen which gave a defined and limited boundary to the cultivated country, and added, at the same time, the pleasing idea that it was sequestered and solitary. The sea coast corresponded in variety and beauty with the inland view. In some places it rose into tall rocks, frequently crowned with the ruins of old buildings, towers or beacons, which, according to tradition, were placed within sight of each other, that, in times of invasion or civil war, they might communicate by signals for mutual defence and protection. In other places the shore was of a more gentle description, indented with small bays where the land sloped smoothly down, or sent into the sea promontories covered with wood. DICTATION (2). made the soldiers point out her bearings so exactly that when darkness came he ran his boat straight on board, and the intrepid officer stood in safety on the quarter deck. ENGLISH COMPOSITION. "Nothing venture, nothing have;" An Easter Holiday. You are expected to write about a page. ENGLISH GRAMMAR. Time allowed, 14 hours. 1. Small service is true service while it lasts; (a) Parse the words printed in italics in the above verse. (b) Say how many distinct sentences there are in the verse, and what is the subject and the predicate of each. After waiting some days, Captain Grant made another attempt to escape from France, and prevailed upon a boatman, by a promise of ten pounds, to row him in the night towards a small island, where, by custom, the English vessels watered without molestation, and in return permitted the few inhabitants to fish and traffic without interruption. In the night the boat sailed; the masts of the British ships were dimly 2. Say what you mean by the words relative, pos (c) Write out in your own words the meaning of the verse. sessive, imperative, infinitive, and objective, and form sentences containing examples of each. In these words, tell the use and meaning of the syllables printed in italics, and give some other examples of words formed in the same way: Unfair. Sweeten. Improper. Snowy. Faithful. Strongly. Clearness. Princess. Hopeless. Oxen. 4. If there is anything wrong in the following sentences correct them, and give your reason for doing so: seen on the other side of the island; and the ter- This is the best of the two. He invited I and you to the house. If everyone swept before their own doors, I did not think it was him. 5. Make a list of adverbs, and, if you can put them into classes. Mention any adverb which is capable of being used as a preposition, and give sentences showing its use. 6. Mention different ways in which the past tense of verbs may be formed. Give the past tense of the verbs: Sit, Lay, Lie, Raise, Go, Am, and Let. (a) is service it lasts scorn not that it casts ANSWERS. The third person singular of the verb To be, indicative mood, present tense. A common noun, third person, singular number, and nominative case following the verb is. A neuter pronoun, 3rd person singular number and nominative case to lasts. A regular intransitive verb, indicative mood, present tense, third person singular, having it as its nominative case. A regular transitive verb, imperative mood, present tense, second person singular, having you understood as its nominative. A negative adverb. A relative pronoun, third person singular number, having shadow its antecedent and objective case, governed by casts. A neuter pronoun, third person singular number, nominative to casts. A regular transitive verb, third person singular number, indicative mood, present tense, having it for its nominative case. Protects A regular transitive verb, indicative mood, present tense, third person singular, agreeing with daisy as its nominative. We should not despise an humble friend: for small service is real service while it continues, just as a daisy by its tiny shadow protects the dewdrops from the sun. 2. A relative usually refers to some word or phrase going before, which is called its antecedentThe man who practices temperance acts prudently. A possessive always denotes ownership or possession-This book is mine, but that is William's. Imperative" is used for commanding, exhorting, or entreating."-Murray. Obey the laws of thy country. Infinitive simply "expresses action or being unlimited by number or person."-Hazen. To learn Greek requires great perseverance. Object is the noun or phrase to which the action of a transitive verb refers-The student learns Greek. un means not, as in unsafe, unkind, undesirable. en means to make, as in tighten, shorten, brighten. im means not, as in imprudent, impure. y means abounding in, as in stormy, woody. ful means full of, as in wakeful, painful, cheerful. ly means like, as in favourably, plainly. ness a prominent quality, as in cheerfulness, greatness. ess a feminine ending, as in duchess, abbess. less without, as in helpless, childless, eyeless. en old plural form, as in oxen, eyen. 4. "Best" should be "better," the comparative being always used when two things are referred to. "I" should be "me," the objective after the transitive verb invited. being, Of whom are you speaking? "Who" ought to be "whom," the construction "Their" should be "his own door," because every is singular. "Him" should be "he," because was takes the same place after as before it. 5. (1) Manner (2) Time (3) Place (4) Number (5) Cause (6) Quantity. much, very, fully, how. (7) Affirmation. yes, truly, doubtless. (8) Negation no, not, nowise. haply, perchance. down he was knocked down; he ran down the : street. 6. (a) 1. By the addition of d or ed to the present. 2. By making no change whatever. 3. By a change in the root only. 4. By a change in the root and a suffix. Sit, sat; lay, laid; lie, lay and lied; raise, raised; go, went; am, was; let, let. FRENCH. Time allowed 14 hours. [N.B.-The French hardly requires answering from us. We shall be prepared to reply to any questions on it in the "Correspondence" column.] Translate into English: I. La plus terrible des maladies, la peste, avait attaqué les Animaux, et faisait parmi eux de grands ravages. Tous étaient frappés, et an grand nombre périssait tous les jours. Dans cette calamité, le Lion convoqua une assemblée générale, et y prononça ce discours: Mes chers sujets, je crois que le ciel a envoyé ce fléau pour punir nos crimes. Que le plus coupable d'entre nous se dévoue pour le salut commun. Le courroux du ciel a souvent été apaisé par de pareils sacrifices. Ne nous flattons donc point, et que chacun examine ici sa conscience. Pour moi, afin de satisfaire ma gourmandise, j'ai mangé un grand nombre de Moutons. Quel mal m'avaient-ils fait? Aucun. Même, il m'est quelquefois arrivé de dévorer le Berger. Je me sacrifierai donc, s'il le faut. Mais je pense que chacun, doit s'accuser aussi, car il est juste que le plus coupable périsse. Sire, dit le Renard, votre Majesté est trop scrupuleuse. Hé bien! Manger quelques Moutons, canaille stupide et méprisable, est-ce là un péché? Non, non. Vous leur avez fait bien de l'honneur en les avalant. Quant au Berger, vous n'avez fait qu'user du droit de la guerre. Ainsi parla le Renard, et son discours fut applaudi. II. La barque passait fort près de l'île lorsque l'enfant était tombé. Daniel nageait comme un poisson, de sorte qu'en deux minutes il fut à l'endroit où le pauvre petit se débattait et commençait à enfoncer. Il le saisit d'une main et continua à nager de l'autre, jusqu'à ce que la barque, que le courant avait entraînée plus loin, pût ravenir à leur secours. On les sortit de l'eau tous les deux. Le beau petit garçon, que l'on nommait Georges, n'avait pas même perdu connaissance. Sa mère, ivre de joie, ne pouvait se lasser de remercier Daniel de lui avoir sauvé son fils. Elle lui demanda comment il se faisait qu'étant si jeune, il sût déjà si bien nager, comment il se trouvait dans cette île, et lui fit tant d'autres questions qu'il fut obligé de lui raconter toute son histoire. Lorsqu'elle eut appris qu'il n'avait plus ses parents, elle lui dit: "Mon cher enfant, tu vas venir à notre château, pui est tout près d'ici, pour changer tes vêtements et te sécher." 2. Give the first person singular of the pluperfect, indicative, and subjunctive present of the following verbs: Avoir, marcher, recevoir, vendre, dormir, mettre. 3. Write down third person singular of the past definite, present subjunctive, and imperfect subjunctive of the following verbs: Battre, boire, écrire, naître, savoir. 4. Translate the following expressions into French: Look at that man. Bring him to me. Bring me a book. The dog died yesterday. will leave at half-past six. He GEOGRAPHY. Time allowed, 14 hours. [Extra marks will be given for neatness in mapdrawing, but only so far as it is accompanied by aceuracy.] 1. Explain why the number of miles in a degree differ, or are alike, according as we measure, degrees of Latitude or Longitude. 2. Name the five oceans and the five continents in the order of the respective size or area. 3. Explain the words Delta, Sierra, Pampas, Prairies, Atoll, Moraine, naming positions or places where they occur. 4. Describe the course taken in going by rail from London to one of the principal sea-ports of England, naming three towns upon the line. 5. Draw an outline map of Ireland, dividing it into provinces, and sub-dividing the N.E. province into counties. 6. Mention the principal mining, manufacturing, and grazing districts of Great Britain, specifying the sub-divisions of the mining districts. 7. State in one column where the following are, and in a second column why they are well known Trafalgar, Valparaiso, Stratford-onAvon, Moscow, Quebec, Nevil's Cross, Bombay, Alexandria, Ballarat, Richmond (U.S.) 8. Mention the chief towns, river mouths, islands, headlands, and bays passed on a coasting voyage from Calais to Marseilles. paid to orthography, handwriting, punctuation, N.B. In all your exercises attention should be grammar, and correctness of expression. ANSWERS. 1. The degrees of latitude are all alike, because they are each the three-hundred-and-sixtieth part of equal circles, namely the meridians; those of longitude being the 360th of unequal circles, namely, the parallels, are consequently unequal. 2. Pacific, Atlantic, Indian, Arctic, Antarctic; Asia, America, Africa, Europe, Australia. 3. The term delta, from the 4th letter of the Greek Alphabet, is a triangular-shaped land formed at the mouths of some rivers by the deposit of alluvial matter carried down by the stream. There is one at the mouth of the Nile. Sierra, means "saw," and is applied to the tops of mountains in Spain, Portugal, Mexico, and S. America. Pampas, are flats or plains near the river La Plata, in S. America. Prairies, are treeless plains in the Western States of N. America. Atoll, a small island with a lake in its centre. They are numerous in Polynesia. Moraine is the debris from an avalanche. There are several in the Alps. |