8. The reflective pronouns in the imperative of reflective verbs | he intend to go to France? 26. He intends to go to France in follow Rule 4 of Sect. XXVI., and Rules 3, 4 of Sect. XXVII., one month. 27. Is your sister to leave to-morrow morning? and also § 100 (2) (3). 28. She is to leave to-day if (s'il) it is fine weather. 29. What do people say of this ? 30. Nothing is said about it Assey ons-nous; asseyez-vous, Let us sit down; sit down. RÉSUMÉ OF EXAMPLES. À quoi vous appliquez-vous ? Il se porte passablement bien. Vos amis se promènent-ils à cheval? tard. Combien ce drap se vend-il le* yard? To what do you apply yourself? I occupy myself with my affairs. Do you apply to your father? How is your father ? He is tolerably well. Do your friends take a ride to-day? Do you like walking (for pleasure) ? Let us not sit down, it is too late. How much is that cloth sold a yard? 1. Comment ce Monsieur s'appelle-t-il? 2. Je ne sais comment il s'appelle. 3. Cette dame ne s'appelle-t-elle pas L. ? 4. Non, Madame, elle s'appelle M. 5. Monsieur votre père se porte-t-il bien ce matin ? 6. Il se porte beaucoup mieux. 7. Fait-il beau temps aujourd'hui ? 8. Il fait un temps magnifique, n'allez-vous pas vous promener? 9. Nous n'avons ni cheval ni voiture. 10. Ne pouvez-vous marcher? 11. Je suis trop fatigué pour marcher. 12. N'allez-vous pas à cheval tous les matins ? 13. Je me promène tous les matins. 14. Comment vous promenez-vous ? 15. Quelquefois à pied et quelquefois en voiture. 16. A qui vous adressez-vous quand vous avez besoin d'argent? 17. Je m'adresse à mon banquier? 18. Ne voulez-vous pas vous asseoir ? 19. Nous vous sommes bien obligés. 20. Ce drap se vend-il fort bien? 21. Il se vend fort cher. 22. Ne devez-vous pas aller à la campagne, s'il fait beau temps? 23. Votre frère doit-il quitter la ville aujourd'hui ? 24. Il doit partir demain matin. EXERCISE 66. 1. Does your sister walk every day? 2. She takes a walk every morning. 3. She likes riding on horseback and in a carriage. 4. What is that little girl called? 5. She is called L. 6. Is not that gentleman called L. ? 7. No, Sir, he is called G., and his cousin is called H. 8. How is your brother? 9. My brother is very well, but my sister is not well. 10. How are your two daughters? 11. They are tolerably well to-day. 12. Will you not sit down, gentlemen? 13. We are much obliged to you, Madam, we have not time. 14. Does that book sell well? 15. It sells very well. 16. How is that silk sold an ell (l'aune) ? 17. It is sold at six francs n 18. Is it fine weather to-day? 19. It is very fine we not take a walk ? 20. I have no time to wall brother apply? 22. He applies t brother at home? 24. No, Sir, he The English a or an before a me the sticle le or la, etc. 5. Se coucher corresponds to the English verbs to retire, to go to bed. I retire early. Je me couche de bonne heure, I rise at the break of day. Vous coupez-vous les ongles ? Vous coupez-vous les doigts ? au coucher du soleil. Nous nous levons de meilleure heure que vous-au lever du soleil. Associé, m., partner. Do you cut your nails? Do you cut your fingers? I often cut my fingers, when I mend my pon. Do you remember the misfortunes of I remember his misfortunes. I remember having seen him. He rises at five o'clock in the morn- VOCABULARY. De meilleure heure, Feu, m., fire. Charpentier, m., car- Main, f., hand. penter. Se chauff-er, 1, ref., to 1. Le perruquier se S'occuper, 1, to occupy Perruquier, m., hair- coupe-t-il le pouce? 2. Non, Monsieur, il se coupe les cheveux. 3. Le charpentier ne se coupe-t-il pas la main? 4. Il ne se coupe pas la main, il coupe le bois. 5. Ne vous rappelez-vous pas cette dame ? 6. Je me rappelle cette dame et ces messieurs. 7. De quoi vous occupez-vous? 8. Nous nous occupons de nos affaires. 9. Vous souvenez-vous des nes your fusils qu'a votre père? 10. Je ne m'en souviens point du tout. Is his 11. Cette petite fille ne se brûle-t-elle pas ? 12. Elle ne se does brûle pas, il n'y a pas de feu dans le poêle. 13. Pourquoi le boucher ne se chauffe-t-il pas ? 14. Parcequ'il n'a pas froid. h by 15. Ces enfants se lèvent-ils de meilleure heure que moi? 16. Ils se couchent de bonne heure, et ils se lèvent tous les matins à 1. Do you rise early when you are well? 2. When I am well I rise every morning at five o'clock. 3. Do you remember your cousin L. ? 4. I remember him perfectly well. 5. Do you go to bed early? 6. We go to bed at ten o'clock. 7. Does not the tailor burn his fingers? 8. He does not burn his fingers, his iron is not warm. 9. Does the carpenter cut his thumb ? 10. He cuts neither his thumb nor his hand. 11. Why do you not warm yourself? 12. I do not warm myself, because I am not cold. 13. Is it not very cold to-day? 14. It is not cold to-day, it rains. 15. Does your hairdresser rise at sunrise ? 16. The carpenter rises at sunrise and goes to bed at sunset. 17. Do you rise earlier than I? 18. We rise every morning at the break of day. 19. Do you cut your hair often? 20. I cut my hair and my nails every month. 21. Do you remember that gentleman ? 22. I remember him very well. 23. I do not remember him. 24. Do you cut your fingers when you mend a pen? 25. I cut my hand when I work. 26. Do you remember what you learn? 27. I do not remember all that (tout ce que) I learn. 28. Do you know if your father is well? 29. He is very well to-day. 30. Is not your mother well? 31. She is not very well. SECTION XXXVII.-USES OF SOME REFLECTIVE VERBS. ville. Quand, when. 1. Aimez-vous à demeurer à la campagne ? 2. Je préfère la campagne à la ville. 3. Vous ennuyez-vous souvent à la campagne ? 4. Quand je m'ennuie à la campagne, je reviens à la 5. Reçoit-on des nouvelles du Général L.? 6. On n'entend pas parler de lui. 7. Vous trompez-vous quelquefois ? 8. Tout le monde se trompe quelquefois. 9. Le banquier trompe-t-il ses clients? 10. Il ne trompe ni ses clients ni ses 11. Ne vous trompez-vous pas amis, il ne trompe personne. dans ce mémoire? 12. Je ne me trompe pas. 13. Vous amusezvous à lire ou à écrire? 14. Je m'amuse à apprendre l'allemand et le français. 15. Avez-vous tort d'apprendre les langues? 16. J'ai raison de les apprendre. 17. Vous ennuyez-vous souvent? 18. Je m'ennuie quand je n'ai rien à faire. 19. À quoi vous amusez-vous quand vous êtes à la campagne? 20. Nous nous promenons le matin, et nous travaillons le reste de la journée. EXERCISE 70. 1. Are you not mistaken? 2. I am not mistaken. 3. Is not the banker mistaken? 4. He is not mistaken, but his clerk is certainly mistaken. 5. Does he not deceive you? 6. He does not deceive me, he deceives nobody. 7. Are you not wrong to deceive your father? 8. I do not intend to deceive him. 1. The verb tromper, conjugated actively, corresponds to the 9. Does not the merchant make a mistake? 10. He makes a English verb to deceive. mistake in the bill which he writes. 11. Do you like the country or the city? 12. I prefer the city; I soon become weary of the country. 13. Does not that child weary you by his ques14. Does not that long story weary you? 15. It does not weary me, it amuses me. 16. Do you amuse yourself when you are in the country? 17. I amuse myself; I learn French tions ? 3. Ennuyer [§ 49 (2)], used actively, means to weary the mind, and Italian. 18. Are you not weary of remaining at your to tease, to bore. uncle's ? 19. I am never weary of remaining there. 20. Is RECREATIVE NATURAL HISTORY. THE MOLE. Is this a well-known animal? A countryman will smile at the 5. Je m'ennuie means, in fact, I am mentally weary, I want question; he knows full well the mole-hills which obstruct his change, amusement, occupation, etc. Je m'ennuie partout, I find no amusement anywhere. 6. S'amuser answers to the English expressions to amuse one's self, to take pleasure in, to spend one's time in, to find amusement in, to enjoy one's self. Nous nous amusons à la campagne? RÉSUMÉ OF On se trompe souvent soi-même en vous trompez-vous pas fréquemment ? We enjoy ourselves in the country. EXAMPLES. scythe in badly-kept meadows, and has often seen the dark culprit gibbeted on the top of a cleft stick. But how many people who will answer this question, and we therefore promise never in London have seen a mole? There is no Registrar-General again to propose such a query. Is the mole clever or stupid? What do the majority of our readers say? Some declare that "the little gentleman in velvet" is a decided genius, and his less enthusiastic friends claim for him a considerable degree of respect. The creature has the repute of being a most skilful engineer, We often deceive ourselves while in which he is a self-taught and natural genius; yet so modest seeking to deceive others. Is not your clerk mistaken? that his finest works are hidden from observation. He never has any money, yet always wears a beautiful coat, for which no thanks are due to any tailor in Great Britain. The mole is, though small, a great eater, hard work giving him a capital appetite; yet he generally contrives to provide very good dinners at all seasons of the year. The teetotalers speak of him with respect, though none of his children belong to the "Band of Hope," nor has he ever worn the temperance medal. His You weary your friends by your love for water-drinking amounts to a passion, but this is per RECREATIVE NATURAL HISTORY. & fiery temper, and much pugnacity of spirit, but even these people admit that he is a good husband and devoted father. 335 Nimrod. Readers will thus see that all the roads are connected with the central house, and form one combined system of animal engineering. One "highway" runs straight from the fortress to the extremity of the hunting ground, and in this the traps are set by the experienced mole-catchers. Most of these galleries are just large enough to allow the animal to pass, and the speed with which he can gallop through such close tunnels is amazing. Experiments were made on this point by a French gentleman, named Le Court, who devoted many years to the study of the mole's habits. He often frightened the creature when feeding by sending the blast of a trumpet into its diningOf course the horrified quadruped set off at full speed towards its citadel, and those who observed the experiments declare that the pace was equal to the swift trot of a horse. How could the speed be ascertained when the animal was hidden? Le Court and his helpers, having ascertained the direction of the "highway," inserted bits of straw into the long passage while the mole was out feeding, and at the top of each straw was a small paper flag. As the startled creature dashed along the gallery each straw was of course forced aside, and the corresponding motions of the paper banners indicated the pace of the little racer. room. Such are some of the qualities ascribed to the mole, and we will now take the liberty of looking a little into his mode of life, that we may see whether the truth has been told about him. But first, a word or two respecting his names, which are three in Great Britain, mole, want, and mouldiewarp. The second of these seems to have been derived from the old Danish wand, and the third from two Saxon words signifying a "thrower up of mould." Learned men, of course, call our little friend talpa, and he is thus designated in natural history. Is the mole a true-born Briton ? We venture to answer "Yes." There is a Cornish legend, telling how the first mole came into existence, and we must not venture entirely to pass over this wonderful history. The story must be well known to some readers, but these may not object to a repetition, which will bring the tradition to the knowledge of others. Be it then known to all, that many ages ago there lived in Cornwall a beautiful damsel named Gwenda. She was fair, as became a Briton, tall, and gifted with a pair of blue eyes of the soft, loving, and poetic type. Every unmarried gentleman in Cornwall wished to make her happy," of course. But Gwenda was so proud of her beauty that she scorned all advances, and for a long time loved her own sweet self only. But at last the hour of her fate came; she fell in love with a famous knight, by name Sir Aymeric. Will our readers believe the astounding statement, that the gentleman did not return the love? Perhaps he was looking out for "a good match," and preferred a long purse to beauty; perhaps he wanted a learned lady, who would study with him in winter evenings the poems of the bards and the philosophy of the Druids; perhaps he wanted a musical lady, who would sing soothing ballads to the knight when out of temper through indigestion, or worried with politics; perhaps but we give up guessing. The simple fact was, that the unwilling Sir Aymeric did not return the love of Gwenda. She, however, determined to conquer his obdurate heart. Her sole trust was, not in her sense, her education, or goodness, but in her beauty. A great feast was to be given on a certain day by the Prince of Cornwall, at Tintagel, to which Gwenda, her mother, and Sir Aymeric were invited. . The damsel procured the "most lovely" dress which Cornish taste could design, and, thus armed for heart conquest, took a last look at her mirror before leaving for the feast. She uttered one exulting exclamation of certain triumph, her mother heard, and "hoped" her beautiful daughter might succeed. “I am sure to conquer," was the bold and self-confident answer of the haughty lady. What followed? A piercing scream was heard; the startled mother looked round, and lo! Gwenda had vanished. They "sought for her high and they sought for her low," but Gwenda was never seen again. All Cornwall was in a panic; other fair ladies might disappear in the same unaccount-stant injury by friction against rough ground. See, then, how able manner; it was really a serious matter for mothers, damsels, and lovers. Was no clue ever discovered? One day, while the old gardener was at work, he picked up a richly jewelled ring, which he knew had belonged to the long-lost Gwenda, and which she had worn on the night of her disappearance. The ring was discovered close to a hillock from which the gardener often saw a mole emerge, and then run up and down the garden path with a strangely melancholy cry. A "wise woman called in; she watched for the appearance of the mole, and then declared, in mystic words, that Alice had been turned into that very mole, as a punishment for her pride, by the mighty spirits of Fairy Land. Such was the origin of the first mole in Cornwall, and this became the mother of all the moles in England. The legend does not inform us where the first gentleman mole came from, and we must leave this matter in a teazing obscurity. was Let us now consider the mole's works and ways. Has the reader ever seen one of this animal's castles? If not, he knows little of the creature's engineering skill. Its fortress resembles some of those ancient camps found in various parts of England, where a central stronghold is surrounded by two or three circles of earthwork. The innermost home of the mole may be called his citadel; round this runs the first circular gallery, bored through the earth by the active engineer. The central house is connected with this first gallery by three roads running from the citadel. Round the first circle stretches a second, and to this four or five roads run from the first gallery. From the second circular work seven or eight tunnels extend far under ground, opening up a large hunting domain to the subterranean The mole, we may well suppose, has a nursery, which he does not place in the citadel, but at a distance, where a special apartment is formed for the education of his five or six babies. When the infants are able to run about there is plenty of playground, the nest being usually at a spot where three or four roads cross. Most readers will now admit the mole to be an engineer; but they may ask, where are his tools? He always goes about with a patent pickaxe, a shovel, and a boring machine, so beautifully made that the most famous engineers have never been able to equal them. These tools are all combined in one piece, and the reader may see them whenever he takes up a mole. Look at the two fore feet, how like hands they are; see how they are turned sideways, so that as the earth is scooped out it is all flung behind, not to impede the animal's work in front. Notice, in the next place, what an admirable miner's dress the mole wears, and how suited it is to his work! As the passages through which he moves are but just the size of his body, rapid motion would be hindered if the earth stuck to the fur. But no soil can cling to a coat which has the softness and smoothness of the finest velvet, combined with a peculiar surface, repelling the most adhesive mould. The mode in which the fur is inserted in the skin is worthy of notice. Each hair grows from the skin in such a direction that the fur will lie even and close, whether rubbed forwards or backwards, without irritating the mole. But while the fur is thus soft and yielding, the skin itself is hard, and so tough that a very sharp knife is needed to cut through it. A tender skin would have been liable to con well the little creature is fitted for his work as an underground. engineer. A hand or foot adapted for boring, scooping, and shovelling back the earth; while the fur and skin are beautifully fitted for subterranean operations. Has the mole eyes? How do our readers answer the question? The ancient Greeks, Romans, and many moderns, have replied by a "No." Let the reader examine for himself. He will find two little, black, shining points deeply fixed in the head, and almost hidden by the fur. These are the eyes. But can the creature really see by these minute organs? The same question occurred to Le Court, and he answered in the proper way by an experiment. Some moles were placed in disused water-pipes, open at the end. If none of the observers stirred, the animals soon made their escape, but if even a finger was put before the opening, they instantly retreated. This seemed to prove the existence of vision. How, then, could such a naturalist as Aristotle deny the possession of sight to the animal? explanation is easy. There is a species of mole in the South of Europe which has the eyelids quite closed, and which is of course blind. The ancients were probably acquainted with this mole only, and many of the moderns çarelessly applied the observations of the old writers to all kinds of moles. Shakespeare describes the animal as "blind," following the notions of his age. But what can a subterranean worker want eyes for ? Perhaps one use may be to give notice to the animal of its approach to the surface, the first gleam of light warning the mole back to deeper recesses. Sometimes, too, our undergrow labourer does leave his dark caves for a moonlight hy The object being to catch and eat as many fat snails as he can find. It is during these nightly rambles that the mole is sometimes snapped up by a hungry owl, in want of a supper for herself and ravenous family. The owl and owlets have, probably, little cause for rejoicing; a severe fit of indigestion must surely be their fate after swallowing the tough skin of the mole. This animal is a great eater; in what food does he most delight? Earth-worms form the daintiest dinners of the hungry little fellow. But he is a bit of an epicure, objecting to eat the worms until they have been skinned. He is said to perform this operation for himself in the neatest manner. Those who are acquainted with the structure of the earth-worm will not be surprised at the mole's objection to the skin. Even a hungry man would object to a mutton chop with 120 bits of gristle in it. The earth-worm has that number of gristly rings in its body, and the epicure mole is therefore quite justified in separating them with the skin. Of course it is very bad for the worm, but then it is very good for the mole. fights. How so? Surely his name would not make a good party cry; he could never aid in blinding the eyes of voters, nor is he valuable enough for a bribe. Will the reader be kind enough to imagine himself present at a dinner party of Jacobite gentlemen in the reign of Queen Anne? What toast is that which excites such uproarious applause? They all drink it; every glass is emptied at the words, "Here's to the little gentleman in velvet." What can be meant? One of the party explains that "the little gentleman" refers to the mole which raised the hill against which the horse of William III. stumbled, breaking the royal rider's collar-bone, and thus causing the death of the king. The delighted Jacobites expressed their frantic joy by thus toasting the unknown and unconscious mole. What if the very animal which raised that hillock had been discovered? Surely he would have been feasted on the fattest earth-worms off silver plates while living, would have been embalmed when dead, and preserved as a precious mummy in a golden shrine with rushlights burning round it day and night. SECTIONAL VIEW OF A MOLE'S NEST AND THE SUBTERRANEAN GALLERIES AND TUNNELS SURROUNDING IT. We have called this quadruped a teetotaler, for in respect to water, the little fellow may well be called "a thirsty soul." So incessant is the desire to drink, that it actually constructs a series of tanks for collecting and holding water, unless a stream or pond be near. A mole has what may be called "a bit of a temper," and will fight most desperate battles with its own kind. Especially does this occur when one happens to bore into the gallery of another. The two pugnacious engineers meet; there is no room for passing; perhaps neither will go back; all the dignity of mole nature forbids that; and there is nothing left but "a set to." This is no joking matter to either of the warriors, for the mole's bite is like that of a bull-dog, as any reader may test for himself whenever he catches one of our black little friends alive. Our mole has the character of being an affectionate husband; in truth, many of his most furious battles are fought in defence of wife and babies. He will often die rather than desert his spouse. The lady mole is sometimes caught in traps, and the devoted husband has been known to perish rather than abandon her. What a nice text this would be for a sermon addressed to certain bipeds! Politicians have sometimes used the mole in their partisan Here some may turn from politics to more prosy matters, by asking whether the mole does not do a great deal of mischief. The farmers certainly bring heavy charges against him, but these accusations may be reduced to two-eating or injuring the roots of crops, and disfiguring the meadows by the numerous "hills" which the busy animal throws up. This last result would be in reality a benefit if the agriculturist would level the hillocks, and thus distribute a surface-dressing of rich soil over his land. The reader will see how small are the evils produced by the mole. A war of extermination has, however, long been waged against our active engineer. In this contest mole-catchers are the fieldmarshals, the artillery consisting of cleverly devised traps, which only moles of first-rate talent are able to avoid. Great has been the slaughter of the quadrupeds; one "catcher" summed up his own slain at 40,000 moles, and even their enthusiastic his torian, Le Court, captured 6,000 in five months. The mole has, we think, made a good fight for life, or not a single one of his race would be now alive. He seems still determined to keep up the "battle of life," and has not given the slightest hint of surrender, or even of emigration. We cannot help honouring such pluck, and wish him all the success he deserves. B A describe a square that shall be equal in superficial area to the squares described on these lines. First draw two straight lines of indefinite length, P Q, R S, intersecting each other at right angles in the point c. On CP and c s set off c D, C E, each equal to A, and on C R, CQ set off c F, C G, each equal to B. Complete the squares C D HE, C F K G, by Problem XVIII. (page 255) and join G E. Upon G E construct the square G E L M, also by Problem XVIII. The square G E L M is equal in superficial area to the squares C D H E, C F K G, described on the given straight lines A and B respectively. LESSONS IN GEOMETRY.-XI. PROBLEM XXVIII.-To draw a triangle of which the base, the sum of its remaining sides, and one of the angles at the base are given. Let the straight line A represent the length of the base of the required triangle, B the sum of its remaining sides, and the angle c one of the angles at its base. Draw any straight line, X Y, of indefinite length, and at any point, D, in it, make the angle Y. D E equal to the given angle c. Then set off DF equal to A along DY, and D G equal to B along D E, and join G F. At the point F in the straight line G F make the angle G F H equal to the angle D G F, pro-ings, such as those of 'ducing FH, if neces- the carpenter, mason, garsary, until it meets the dener, wheelwright, etc., side D G of the triangle in which a knowledge of DGF in the point H. geometry is indispensable, if he who chooses any one of them as the avocation Fig. 38. The triangle H F D is the triangle required, for its base, D F, is equal to the given straight line A, one of its angles F D H is equal to the given angle c, and the length of its remaining sides, D H, H F, taken together, is equal to B, for since the angle HFG is equal to the angle H G F, HF is equal to H G, and DG, or DH+H G, was made equal to B. The position of the point H in the straight line D G may also be found by bisecting F G in K, and drawing K L perpendicular to FG, and cutting D G in H. PROBLEM XXIX.-To draw a triangle having its angles equal to the angles of a given triangle and its perimeter, or the sum of its three sides, equal to a given straight line. с Let the straight line A B represent the length of the perimeter, or sum of the three sides of the required triangle, and C D E the given triangle to whose angles the angles of the required triangle must be equal. At the extremity A of the straight line A B make the angle B A F equal to the angle E D C of the triangle C D E, and at its extremity B, make the angle A B G equal to the angle CED. Bisect the angles B A F, A B G by the straight lines A H, BK, and let these straight lines be produced far enough to intersect in the point L. From the point L draw L M parallel to A F, meeting A B in M, and L N parallel to B G, meeting A B in N. The triangle L M N thus formed is the triangle required, for it is manifest that its angles at L, M, and N, are equal to the angles at C, D, and F of the triangle C D E, for the angle L M N, by Theorem 2 (page 156), is equal to the angle B A F, which was made equal to the angle C D E, and the angle L N M, by the same Theorem, is equal to the angle A B G, which was made equal to the angle C ED; and if there be two triangles each one of which has two angles which are equal to two angles of the other, the remaining angle of the one must be equal to the remaining angle of the other, since the three angles of every triangle, whether great or small, are together equal to 180 degrees; and as in the triangle L M N there are two angles L M N, L N M, equal to the angles C D E, CED of the triangle C D E, the remaining angle M L N of the triangle L M N must be equal to the remaining angle D C E of the triangle C D E. Now the side M L is equal to M A, because the angle M L A is equal to the angle MA L, M L A being equal to L A F or H A F, because they are alternate angles, and HAF being by the construction equal to MA H. For the same reason the side N L of the triangle L M N is equal to N B. Therefore the perimeter of the triangle L M N, or the sum of its sides L M, M N, N L, is equal to the given straight line A B. PROBLEM XXX.—To describe a square that shall be equal in superficial area to the sum of the squares described on two given straight lines. M Fig. 39. Let A and B be the two given straight lines; it is required to VOL. I Now at first sight it is difficult for any one who is endeavouring by self-tuition to acquire a knowledge of practical geometry, whether for an agreeable change from other pursuits and a useful mental exercise, or to aid him in the pursuance of his callingand there are many call H N Fig. 40. A B R by which he must earn his daily bread wishes to rise among his fellows, and so deservedly command the reward of his industry and intelligenceit may, we say, seem at first difficult to perceive that the large square G ELM is exactly equal in superficial area to the two smaller squares CD HE, CF K G, taken together. We will, therefore, first give him the means of proving to his satisfaction, by the aid of his compasses and parallel ruler, that it is so, and then endeavour, as in former cases, to deduce from a consideration of Fig. 40 several geometrical facts that may be gleaned from this problem, without the necessity of treating them in separate problems. And first, for proof positive from ocular demonstration that the area of the large square G E L M is equal to the joint area of the smaller squares C F K G, C D H E. An inspection of the annexed figure, which is drawn on rather a smaller scale than Fig. 40, but in precisely the same proportions, will show the truth of the assertion. The two larger squares are divided into their component parts in the following manner. Through c draw c T parallel tó G M or E L, meeting E G in T, in order to fix the point T. Then through T draw T U parallel to c E, and T v parallel to C G. Along T U set off To equal to c G, and through o draw o x parallel to T v orc G, meeting L M in X, and through v draw v w, parallel to T U or C E, meeting o x in w. Next, for the necessary division of the square C D H E, through c draw c z, parallel to E G, and produce L E, to meet the straight line D H in the point Y. If this figure be drawn on a piece of paper, and the squares C F K G, CDHE be cut out and divided, and the pieces put together on the square G E L M, so that the pieces numbered 1, 2, 3, 4, 5, in the smaller squares, be placed on the divisions similarly numbered in the large square, it will be found that the area of the large square is exactly equal to the joint area of the smaller squares. D 2 C 3 K It will be noticed that the straight lines P Q, R S in Fig. 40 were drawn at right angles to each other, and that the straight lines c G, CE, that were set off along CQ, Cs are at right angles to each other necessarily. This is the point in the construction on which the solution of the problem depends, whatever may be the length of A and B, and to effect it we have only to draw a line equal to A, and at right angles to one end make a line equal to B, and join the extremities of th 22 |