point f; from f, with the radius fg, draw the arc gi; bgi will be the curve required. The curve called the Cyma Recta (Fig. 59).-Let the curve be formed between the lines a b and c d; draw the line bd, and divide it into five equal parts; mark the second division from b, viz., e; upon be describe the equilateral triangle b ef, and upon ed describe the equilateral triangle de g; from f, with the radius fb, draw the arc be, and from g, with the radius g e, draw the arc e d; bed will be the curve required. The pupil can draw an equilateral triangle upon a given line by the following method. Let ab (Fig. 60) be the line upon which the triangle is to be described; from a and b as centres, with the radius ba, describe two arcs intersecting each other in the point c; join c a and c b; the triangle a b c is an equilateral triangle. (See Lessons in Geometry, VII., page 209.) The curve called the Ogee (Fig. 61).-Let it be drawn between the lines a b and c d; draw d e perpendicular to c d, and divide it into four equal parts; through the first from e-namely, hdraw the line hig parallel to a b, make h i equal to he; draw the line ki parallel to e d, and from i, with the radius i k, of a little help from geometry; we advise him also to draw all these lines of arrangement with a light hand, that they may be more easily effaced when done with. To draw the pear (Fig. 63), we will first draw a line to represent the length or axis, and from this line "offsets on each side as shown by dotted lines. The pupil may please himself as to the number of these "offsets" and their whereabouts; he will not be long before he finds that such lines are best arranged opposite, and to meet, angles, and the greatest distance of curvature from the axis. He will then proceed to draw the outline through the extremities of these offsets, especially observing the kind of line requisite between each point: in some parts the outline is more outwardly curved than in others, in some it is nearly straight, in others the curve is inward. If the pupil will exercise his observation in this way when looking at solids and natural objects, which he can do at all times, whether he has a pencil in his hand or not, even when out for a walk, he will be not a little surprised, should he make this his general practice, to find how rapidly he will gain confidence and power, and be able to produce truthful draw the semicircle k gl: join l d, and upon it draw the equilateral triangle 1 m d; from m as centre, with the distance dor ml as radius, draw the arc d n l; the line d n gk will be the curve required. By recommending the practice of geometrical drawing, we only wish to direct the pupil where to find further assistance in free-hand drawing; we will now show, by a few examples, how these principles may be applied. An oval or egg-shaped figure (Fig. 62) would be very difficult to draw, if the boundary line only were to be attempted without some assistance from geometry; there would be a great deal of rubbing out and alteration before it was finished. Let the pupil try the figure in the following manner: first by the help of compasses, then by hand only. Draw the straight line a b, and divide it into two equal parts in the point d. Through d draw ede at right angles to a b, and make d c equal to a d or d b. Construct upon a b the equilateral triangle a e b, and take the point g at one-third of the distance from e to b, and determine the point f in the same way. Then from the points f, g, draw the lines fi, g h, perpendicular to a e and eb respectively, and make each of them equal to one-half of e f or eg. After this arrangement has been made, draw the semicircle a c b and the arca be and a e through h and i. It will be necessary to repeat it a few times, when the pupil will begin to see the advantage The and useful drawings. We will give him another example (Fig. 64), for which he must arrange the scaffolding himself, with one exception, because it includes a principle which we will merely allude to now, as we shall have better and more frequent opportunities by-and-by to enlarge upon it. exceptional assistance we offer in this case, is that of the dotted line which runs through the centre of the handle of the trowel, and passes in a direct course to the point of the blade. We may here observe that an implement of this kind, to be really useful, ought to be so constructed; and if we look at it with an artistic eye, the composition of lines which make up this very simple subject, must strike any one as being more symmetrical than if the handle and the blade had been united at any other angle. This remark upon so insignificant an object as a garden trowel may appear trivial, but it is the principle we contend for, and which is, in reality, of the greatest importance. It is true we might have selected a more noble object, but it would not have better illustrated our meaning, or have made it more evident, and at the same time have provided the pupil with an example for his practice more suited to the experience he has at present attained as a draughtsman. Nature teaches us this lesson, and it is evident everywhere that harmony of line and proportion always accompany the greatest utility and strength. LESSONS IN ARITHMETIC.-XV. DECIMALS (continued). 10. Division of Decimals. CASE 1.-Divide 120-3033 by 3.27. 100 10000 3679 100. 12030333.27 = 1103033 ÷ 327: = 1203033 X 327 3679 is the quotient arising from dividing the dividend by the divisor as if they were whole numbers, and the denominator 100 shows that there must be two decimal places in the quotient. These two decimal places arise, as will be seen by the fraction 100 from the fact of there being two decimal places more in the dividend than in the divisor. CASE 2.-If the number of decimal places in the divisor and dividend were the same, the result would be exactly the same as if the divisor and dividend were whole numbers. Thus, 231 The part of the true quotient already obtained is an integer, the division being in fact the same as that of 35970. If more ciphers be annexed to the dividend, we shall get decimal places in the quotient, and the more we obtain the nearer to the true quotient shall we arrive. 11. These examples will sufficiently illustrate and explain the following Rule for the Division of Decimals. Divide as if the divisor and dividend were whole numbers. If the number of decimal places in the dividend exceed the number in the divisor, cut off from the quotient as many *We shall use the expression true quotient to indicate the total result obtained by the division of one number by another, thus distinguishing it from the quotient defined in Lesson V., Art. 1 (page 69), which is only the integral part arising from a division. decimal places as are equal in number to this excess, prefixing ciphers if necessary. If the number of decimal places in the dividend and divisor be equal, the division will be the same as in whole numbers. If the number of decimal places in the dividend be less than the number in the divisor, annex as many ciphers to the dividend as will make the number equal to the number in the divisor, and then proceed as in whole numbers. 12. We subjoin other examples of division of decimals. EXAMPLE. Divide 1 by 10-473, carrying the quotient to 5 places of decimals. We are at liberty to write 1 thus-1.00000, putting as many ciphers after the decimal point as may be required. Since there are to be 5 decimal places in the quotient, and since there are 3 in the divisor, we must add 8 ciphers. 10:473) 1·00000000 (9548 94257 57430 52365 50650 41892 87580 83784 3796 Hence the required answer is 09548, prefixing a cipher in order to get 5 decimal places in the quotient. 13. EXAMPLE.-Divide 8 by '00002. Annexing 4 ciphers to 8, since there are 5 decimal places in the divisor, we have 00002) 80000 (40000 80000 the division by the rule being, in fact, the same as that of 80000 by 2. 14. It will be observed that we are not required in some cases to find more than a certain number of figures of the quotient when it is a decimal. Sometimes, by continuing the division far enough, we shall find that there is no remainderi.e., that the quotient can exactly be found in the form of a decimal. But if by continually dividing we cannot arrive at a stage where there is no remainder, then we can only get what is termed an approximation to the result. The more figures of the quotient we take, the nearer we shall be to the value of the true quotient. stopped at four decimal places in the quotient, the result would Thus, in the division above performed in Art. 12, if we that 8 is the next figure of the quotient, and therefore this 8 be 0954. Carrying on the operation one step further, we see meaning we are nearer to the true quotient by Where we are required to find a quotient to a given number of places, it is customary to carry on the division to one place more than is actually required, in order to see whether the next figure is greater or less than 5. If it is greater than 5, then we shall be nearer to the true result if we increase the last figure of the required number of places by unity. Thus, in the case above given, finding that the fifth decimal place is 8, the quotient to four decimal places will be more accurately written 0955 than 0954, because 0955-or, what is the same thing, 09550-is nearer to 09548 than 09540 is. Now 09550 is more than 09548; whereas 09540 is 3. How many boxes will it require to pack 71-5 pounds of butter, if you put 5.5 pounds in a box? 4. How many suits of clothes will 29.6 yards of cloth make, allowing 3-7 yards to a suit? 5. If a man can walk 30.25 miles per day, how long will it take him to walk 150.75 miles ? 6. How many loads will 134642-156 pounds of hay make, allowing 1622-2 pounds for a load? 7. If a team can plough 2:3 acres in a day, how long will it take to plough 63-75 acres? 8. How many bales of cotton are there in 56343 75 pounds, allowing 375 pounds to a bale ? 9. Determine the quotient in the following examples in division of decimals by removing the point in such dividend to the left, and adding ciphers when necessary : Emménagement Emménagements Emménager Emménagogue Emmener Emmenotter Emmiellé Emmuseler Anh-ma-re-nay ENGLISH. Hooded. Sea-hardened. To man a ship. Anh-may-lay (2nd syll. long) To entangle. Anh-may-nazh-manh Furnishing a house. Sweetened with honey. To consecrate a bishop. To mortise. Banked with earth. 10. Multiply the following numbers together by removing the Enivrer (and all de- Anh-nee-vray decimal points : · rived from it) Enorgueillir (ditto) Anh-or-guaygl-yeer ENGLISH. Intoxicating. Intoxication. To intoxicate. To render proud. to, to come to, in connection with nouns or pronouns representing 3. Aller trouver, venir trouver, are used in the sense of to go 11. When the occurrence is a periodical or customary one, the article le is prefixed to the day of the week or the time of the day. Il vient nous trouver le Lundi, He comes to us on Mondays. noon. EXAMPLES. LESSONS IN GEOGRAPHY.-VIII. DISCOVERIES OF THE NINETEENTH CENTURY, SIR JOHN Ross, who sailed in the Victory in 1829, on an exafter-pedition to the north, again explored Baffin Bay, Lancaster Sound, and Prince Regent Inlet; discovered land which he called Boothia Felix, from the name of his patron; and explored the coasts of this new country, until he was so hemmed in by the ice, that he could neither advance nor return. The expedition accordingly remained in this condition during the space of four years, the longest period on record of the detention of navigators in the northern regions. While thus detained the members employed their time in making excursions which enlarged our geographical and meteorological knowledge, and added to philosophy the fine discovery of the north magnetic pole. Besides the isthmus and peninsula of Boothia Felix, the Do you go to that lady's house on expedition discovered King William Land, and the western sea I am going to speak to your father. Monday? I intend to go there on Tuesday. I generally go there on Wednesdays. VOCABULARY. Commenc-er, 1, to com mence. EXERCISE 45. 1. Qu'allez-vous faire ? 2. Je vais apprendre mes leçons. 3. N'allez-vous pas écrire à vos connaissances? 4. Je ne vais écrire à personne. 5. Qui vient de vous parler ? 6. L'Irlandais vient de nous parler. 7. Quand l'Ecossaise va-t-elle vous enseigner la musique? 8. Elle va me l'enseigner l'année prochaine. 9. Va-t-elle commencer Mardi ou Mercredi ? 10. Elle ne va commencer ni Mardi ni Mercredi; elle a l'intention de commencer Jeudi, si elle a le temps. 11. Votre compagne va-telle à l'église tous les Dimanches ? 12. Elle y va tous les Dimanches et tous les Mercredis. 13. Qui allez-vous trouver ? 14. Je ne vais trouver personne ? 15. N'avez-vous pas l'intention de venir me trouver demain ? 16. J'ai l'intention d'aller trouver votre teinturier. 17. Envoyez-vous chercher le médecin? 18. Quand je suis malade, je l'envoie chercher. 19. Reste-t-il avec vous toute la journée ? 20. Il ne reste chez moi que quelques minutes. 21. Allez-vous à l'école le matin ? 22. J'y vais le matin et l'après-midi. 23. Y allez-vous tous les jours ? 24. J'y vais tous les jours, excepté le Lundi et le Dimanche. 25. Le Samedi je reste chez nous, et le Dimanche je vais à l'église. EXERCISE 46. 1. What is the Irishman going to do? 2. He is going to teach music. 3. Has he just commenced his work? 4. He has just commenced it. 5. Who has just written to you? 6. The dyer has just written to me. 7. Does your little boy go to church every day? 8. No, Sir, he goes to church on Sundays, and he goes to school every day. 9. Do you go for the physician ? 10. I send for him because my sister is sick. 11. Do you go to my physician or to yours? 12. I go to mine, yours is not at home. 13. Where is he? 14. He is at your father's or at your brother's. 15. Do you intend to send for the physician ? 16. I intend to send for him. 17. Am I right to send for the Scotchman ? 18. You are wrong to send for him. 19. Do you go to your father in the afternoon? 20. I go to him in the morning. 21. Does your brother go to your uncle's every Monday? 22. He goes there every Sunday. 23. Are you going to learn music? 24. My niece is going to learn it, if she has time. 25. Am I going to read or to write? 26. You are going to read to-morrow. 27. Does he go to your house every day? 28. He comes to us every Wednesday. 29. At what hour? 30. At a quarter before nine. 31. Does he come early or late? 32. He comes at a quarter after nine. 33. What do you go for? 34. We go for vegetables, meat, and sugar. 35. We want sugar every morning. called after the same sovereign. As to the north-west passage, he found that this did not exist in Prince Regent Inlet, nor to the south of latitude 70° N.; but Sir John Ross failed in discovering a free passage in the frozen seas of America, by which he could find his way to Behring Strait; in fact, the peninsula which separates Prince Regent Inlet from this northern sea, at the place where the expedition made its principal researches, is not only very narrow, but is chiefly covered with lakes which reduce the isthmus between the two seas to a breadth of three miles. Other expeditions, no less dangerous, and equally difficult, if not more so, had been undertaken by land, with a view of exploring the northern regions of America, and the coast of the Polar Sea, in order to assist in the discovery of the passage so ardently sought for during so many ages. Samuel Hearn, employed by the Hudson Bay Company, in 1771 commenced his expedition at Prince of Wales Fort, and discovered the Coppermine River, which he traced to its embouchure in the Polar Sea. Franklin, in 1820-21, made an expedition by land along the same coast, between the Coppermine River and Cape Turnagain. This adventurous expedition, accomplished amidst a thousand dangers, among which famine was not the least formidable, was highly useful in a geographical point of view. Two years afterwards the same officer undertook another expedition to the north, and explored the country between the Mackenzie River and Cape Back; at the same time Dr. Richardson, one of the party, explored that part between the Mackenzie River and the Coppermine River. The part of the coast left unexplored between the limits of Captain Beechey and Captain Franklin's discoveries, extending to 150 miles, was nearly completed in this respect by Captain Back, and after him by Messrs. Dease and Simpson, so that the northern shores of North America are now geographically known almost throughout their whole extent. Our geographical knowledge of the interior of the continent of North America was greatly increased by some other important expeditions. Lewis and Clarke travelled to the sources of the Missouri among the Rocky Mountains, and reached the Pacific Ocean by descending along the course of the Columbia River. Pike, in exploring the sources of the Mississippi, discovered those of the Arkansas and the Red River. Major Long, James Peak, Messrs. Cass and Schoolcraft, travelled over this vast region, so remarkably studded with lakes and rivers, and belonging partly to Britain and partly to the United States. Mackenzie, in 1789, went from Montreal, and travelling to the north-west, descended along the course of the river which bears his name, and found that its source was in the Slave Lake, and its termination in the Arctic Ocean; he then crossed the chain of the Rocky Mountains, and reached the Pacific. In South America, Baron von Humboldt began his explorations, and accompanied by M. Aimé Bonpland, the celebrated botanist, visited Columbia, now divided into the republics of Venezuela and Ecuador, and the Granadian Confederation, studying during his travels all the phenomena of nature, tracing the geography of the country, measuring the heights of the Andes, examining the craters of volcanoes, delineating on maps the courses of rivers, and, in short, exploring the greater part of this magnificent country, On the river Amazon, he made observations equally curious and important. He proceeded from Peru to Mexico, and made similar observations in the latter country; and he has described his scientific discoveries in these regions in a style both effective and interesting; so that in no portion of the globe have greater advances been made in the knowledge of physics and geography, and of all the sciences connected with them. Botanical geography may, in fact, be said to have originated with Baron von Humboldt. If to this we add that the author of the "Tableaux de la Nature" studied the countries in which he travelled both in an economical and political point of view, his merit as a scientific traveller stands unrivalled. The travels of La Condamine in Peru and on the river Amazon; of Smith and Maw, on the same river; of Messrs. Spix, Martins, and Auguste St. Hilaire, in Brazil; of Don Felix Azara, in Paraguay; of Captains King and Fitzroy, in Patagonia and Tierra del Fuego; of M. Stephenson, in Chili and Peru; of M. Gay, in Chili; and of M. Schomberg, in Guiana, have all contributed to the perfection of our knowledge of the geography, the productions, the geology, and the population of South America. Among these later travellers must be mentioned M. A. d'Orbigny, a learned French geologist, who, in 1826, after a sojourn of seven months at Buenos Ayres, ascended the Parana stone, some of which weigh eighty tons. The great gates are each composed of one single mass; and there are colossal images rudely sculptured, showing that at a very early period there must have been some communication between the Old World and the New. The traveller above mentioned then visited in succession the cities of Cochabamba and of Santa Cruz de la Sierra; courageously penetrated into the province of the Chiquitos, which he surveyed in every direction to the river Paraguay and the Brazilian province of Matto-Grosso; noted the manners of the Guarayos, a tribe still entirely savage; traversed the province of the Moxos, to the north-east of Upper Peru; passed some time in the forests inhabited by the Yuracares Indians; discovered the points of discharge of the Rio Beni and Rio Mamoré, tributaries to the Amazon; returned to Santa Cruz; visited Potosi, the city of inexhaustible mines; and finally sailed for France from the coast of Peru. This remarkable expedition lasted for the space of eight years, and produced valuable results for the geographer, the natural historian, and the geologist. as far as 1,000 miles from its mouth, travelled over the province of Corrientes, and other parts of the Argentine Confederation, visited the hordes of savages which people the Grand-Chaco, and returned to a civilised territory, passing through the provinces of Entre-Rios and Santa-Fe. He then travelled into Patagonia, ascended the Rio Negro, and sojourned eight months in that country amongst the stalwart savages, whose Herculean forms and size had been described with so much exaggeration by Pigaletta, Drake, Sarmiento, Lemaire, Byron, Bougainville, and many other navigators. This intrepid naturalist then proceeded to Chili, having doubled Cape Horn and reached Bolivia, sometimes called Upper Peru, of which he explored the western region, rendered so remarkable by the labours of the ancient Quichuas. He ascended the summits of the Andes, and on his reaching the opposite sides of these amazing heights, beheld a magnificent panorama of snowy peaks, and of immense chains of mountains. He at last reached the vast table-land on which 18 situated the great Lake of Titicaca, 150 miles long, rendered so famous by the Temple of the Sun, built by the Incas, on an island in its centre. At the village of Tiahuanacu, near the banks of this lake, are also to be seen the remains of the stupendous palace erected by the ancient Peruvians. The interior courts, 360 feet square, are built of enormous blocks of 66 GEM OF THE PACIFIC." From the extremity of South America let us pass on to the regions which surround the Antarctic pole. There we see navigators of all nations braving the storms and the icebergs of those seas which are covered with everlasting mists, in order to enrich geography with important observations and discoveries. After the immortal name of Cook, came those of William Smith (1818), of Lieutenant Barnsfield, of the Russian officers Bellinghausen and Lazareff (1819), of Botwell (1820), of Weddell and Palmer (1822), of Biscoe (1830), and of Balleny (1839). It is to these navigators, some commissioned by the government of the nations to which they belonged, and others who were simply whalers or seal-catchers, that we owe the successive discoveries of New South Shetland, the South Orkneys, Palmer Land, Trinity Land, the islands of Peter and Alexander, Enderby Land, Adelié Land, Graham Land, and the islands of Biscoe and Balleny. Three voyages in the southern circumpolar seas-those of Dumont d'Urville, of Captain James Clarke Ross, and of the The American Commodore Wilkes-deserve particular notice. French expedition, under the command of Captain Dumont d'Urville, after a careful exploration of the Strait of Magellan, proceeded in 1838 towards the icy regions, and was stopped by an iceberg in latitude 64° S. The two vessels endeavoured to overcome the obstacles which opposed their progress, but they |