3. The three sides of a triangle are 21, 20, and 13; what is its perpendicular? 4. The base of a triangle is 5·96, and its altitude 3.81; what is its area? 5. The base of a triangle is 7.37 chains (1 chain = 66 feet), and the altitude 4.98 chains; what is its area? 6. The hypothenuse of a right-angled triangle is 205, and the base 200; required the area? 7. The side of an equilateral triangle is 3·4; what is its area? 8. Suppose the base of an isosceles triangle, whose area is one acre, is 363 feet; what is its altitude? We subjoin another rule for the calculation of the area of a triangle without finding its perpendicular, the three sides being given-From half the sum of the three sides subtract each side separately. Then multiply the half sum by the three remainders successively, and the square root of the product will be the area. EXAMPLE 1.-The three sides of a triangle are 13, 20, and 21; find its area by the above rule. 13 27 2217 EXERCISE 3. 27 a॰ །g།ནཱ་ 1134 PROBLEM II.-To find the radius of a circle inscribed in a given triangle. Rule: Divide twice the area of the triangle by the sum of its three sides. it is required to find the length of DE. Fig. 14. C EXERCISE 5. 1. The side of an equilateral triangle is 10; what is the radius of the inscribed circle ? 2. The two legs of a right-angled triangle are 3 and 4; what is the radius of the inscribed circle ? 3. The three sides of a triangle are 39, 60, and 63; what is the diameter of the inscribed circle in "Geometry.") being given, to find the radii of the circumscribed PROBLEM III.-The side of a regular polygon (see Definitions and inscribed circles. Rule: Divide 360 (the number of dethe number of sides in the polygon; grees in the whole circumference) by the quotient will be the angle at the centre. E Let ABDE (Fig. 15) be a regular hexagon, of which the side A B is known; then by above rule the angle AOB is found. Halve Then this for AO C, and join o c. oc is perpendicular to AB, and bisects it, and the angle ACO is a right angle. Hence in the right angled triangle A co, we Fig. 15. have given the angle AOC = and the angle OAC = 90° - A o C, also the perpen15876 126 Ans. dicular AC = Then (Problem I.) A 0= and OC=AOX sine o A c. EXAMPLE 1.-The side of a regular pentagon (five-sided) is 14 4536 1134 1. The three sides being 13, 14, 15, what is the area? 2. The side of a hexagon is 10; what is its area computed 15 yards; what are the radii of the circumscribed and inscribed which belong more particularly to Trigonome- and co= 12.972 (A o) × 8090 (sine 54°) 10.495, about. PROBLEM I.-Let A B C (Fig. 13) be a tri- 1. The side of an octagon is 138 yards; what are the radii of the circumscribed and inscribed circles respectively? 2. The radius of a circle is 1.84; what will be the length of the side of a heptagon inscribed in it, and of an equilateral triangle described about it ? 3. If 1 inch is the distance between the opposite sides of a decagon, what is the distance between its opposite angles? We now come to the consideration of the relations which exist between the various lines connected with circles; and first of the proportion between the circumference and the diameter. This may be regarded approximately as 22 to 7, but more correctly as 31416 to 1*. Hence if D = diameter, c = circumference and π = 3.1415, Que j'eusse reçu, that I might have Qu'ils reçoivent. Imperfect. Pluperfect. Que tu eusses reçu. Qu'il reçût. Qu'il eût reçu. [received. Que vous vendiez. Qu'ils vendent. Imperfect. COMPOUND TENSES. Past. Que j'aie vendu, that I may have Qu'il ait vendu. Que nous ayons vendu. Que vous ayez vendu. Qu'ils aient vendu. Pluperfect. [sold. KEY TO EXERCISES IN LESSONS IN FRENCH. 1. Is it you, Madam, who have called your servant ? 2. It is not I who live at No. 18? 15. Est-ce là 17. Est-ce vous, 18. Ce n'est pas 1. Est-ce vous, mon ami, qui m'avez averti de mon danger? 2. Ce n'est pas moi qui vous en ai averti. 3. Est-ce eux que vous attendez de jour en jour ? 4. Ce n'est pas eux que nous attendons. 5. Est-ce vous qui avez fait ceci ?, 6. Ce n'est pas nous, c'est vous qui l'avez fait. 7. Était-ce en Angleterre que vous avez acheté ce chapeau? 8. Ce n'était pas en Angleterre, c'était en Allemagne. 9. N'était-ce pas en Russie que vous avez fait connaissance avec lui? 10. Ce n'était pas en Russie, c'était en Italie. 11. Etait-ce vous qui nous appeliez ? 12. Ce n'était pas nous, c'était lui. 13. Ne connaissez-vous pas les deux Polonais qui lisent? 14. Je connais celui qui est près de vous. la dame que vous attendiez? 16. Ce ne l'est pas. Messieurs, qui avez comblé mon frère de bontés ? nous, Monsieur, nous n'avons pas le plaisir de le connaître. 19. Est-ce vous qui avez été blessé au bras? 20. Ce n'est pas moi. 21. N'entendez-vous pas ces deux dames? 22. Je n'entends pas celle qui chante. 23. J'entends celle qui joue, 24. Est-ce vous qui êtes venu chez nous ce matin ? 25. Ce n'est pas moi, j'étais à Londres alors. 26. Est-ce vous, Monsieur, qui nous avez fait ce plaisir? 27. Ce n'est pas moi, c'est ma sœur. 28. Etait-ce votre fils qui désirait à toute force aller à Londres ? 29. Ce n'était pas lui, il est maintenant en Allemagne. 30. Est-ce vous qui écrivites cette lettre ? 31. Nous n'avons pas écrit de lettre. 32. Qui demeure au No. 20? 33. C'est moi. 34. Est-ce nous que vous avez vus. 35. Ce n'est pas vous que j'ai vu. EXERCISE 157 (Vol. III., page 114). 1. Do you know these strangers? 2. Yes, Sir, they are our neighbour's brothers. 3. Are they not Scotch? 4. No, Sir, they are Swiss. 5. Did not some Scotchmen make you a present of this cap? 6. No, Sir, they are Swiss. 7. Is it not your servant who stole wine from you? 8. It is not he; it is his brother. 9. Is it not he who has taken your preserves? 10. It is not he, it is his children. 11. Are not those the children whom you caught stealing your sugar? 12. They are their brothers. 13. Are they not cousins? 14. They are not cousins, they are brothers. 15. What are those silk goods? 16. They are goods which have just been sent to us. 17. Is not Lyons a fine city? 18. It is a large and beautiful city. 19. Is not that the handkerchief which you have lost? 20. I think so. 21. Do not your windows look on the garden? 22. Yes, Sir, they look on the garden. 23. Is it not our wheelwright who has made that wheel? 24. It is not he who has made it. 25. It is our friends who have broken it, and it is the joiner who has made it. OUR HOLIDAY. SKATING. WINTER, like summer, has its own peculiar pastimes, and chief among these is skating. When the ground is hard as rock, and rivers and streams are stiffened in the grasp of Frost-when Que je vendisse, that I might sell. Que j'eusse vendu, that I might all ordinary sports are thus for a time put an end to this exer 80 may be so temperate as not to afford him an opportunity of putting on his skates. To glide swiftly along in the frosty air, skimming more like a bird than a denizen of the earth, and without the ordinary labour of locomotion, is the height of a skater's enjoyment; and no one who practises this art can fail to be physically benefited in proportion to the pleasure he finds in the exercise. Skating, to the unpractised observer, appears an extremely difficult art; nor is it altogether an easy one for the most determined and agile person to acquire. But it is not nearly so difficult as it seems, if a few preliminary qualifications be taken into account. First of all, it is of course necessary to have good legs, and to be not afraid to use them. The lad who shrinks from a long walk, or a good run at the top of his speed, is destitute of the first qualification to become a skater. The next requisite is courage; for the learner, whoever he may be, must make up his mind to encounter many falls before he can become Thirdly, so proficient as to feel himself at home upon the ice. A timid he requires confidence to expedite him in his task. and mistrustful person, who is constantly in the fear and expectation of coming down, will not find his expectation disappointed. But the one who, as soon as he has gained his footing upon the ice, shoots boldly out, within the limits of his instructions, has the elements of success within him, and will reach it with less than half the mishaps of a less venturesome comrade. Our readers are, no doubt, familiar with the nature and construction of a pair of skates. They are usually composed of wood and iron; the foot resting upon the wood, and the iron forming a narrow ledge in the centre underneath, upon which the skater balances himself. Sometimes, however, the wood is dispensed with, the foot resting upon a light iron framework, just sufficient to afford support and security. As the combination of wood and iron is in more general use, we will presume that the learner, at the outset, will provide himself with a pair of this description, and must say a few words to guide him in his choice. The more nearly the woodwork of a skate is adjusted to the shape of the foot, the firmer will be the support it affords to the skater. Those skates are best, therefore, which have the woodwork hollowed out to fit into the "ball" of the foot and to receive the heel. The iron curving upwards should not project far beyond the front of the foot, and it should be continued its full length, so as to come to the extremity of the heel, or even a little beyond it. The under surface of the iron is sometimes grooved or fluked, and this make is often recommended to the learner as affording a better hold upon the ice. But it has this disadvantage, that if he practises much at a time, or becomes an active skater, the groove is apt to fill up with fragments of the ice, and therefore to render the iron very slippery, and to increase his difficulty in maintaining his footing. We therefore recommend the learner to provide himself with the plain, level under surface, unless he can afford to buy both the grooved skate for his earlier practice, and the plain one for his later exercise. An important point in the purchase of the skates is to see to the fastenings. The leather straps by which they are affixed to the feet should be of the strongest material and make, or they will give way when a severe strain is put upon them, and The heel part should be furnished with involve a severe fall. a screw, to go up into the heel of the boot, a hole being made in the latter with a gimlet for the purpose. The skater should always make these holes, or clear them out, before he sets out on his expedition, or he may find it an awkward matter to do so without taking off his boots when he reaches the margin of the ice. The price of a pair of skates varies greatly. Sometimes a good serviceable pair may be bought secondhand for three or four shillings; but the shop price for new skates ranges from seven shillings and sixpence to twice that amount, according For skates of fancy or patent construction to quality, etc. still higher prices are asked. Having obtained his skates, and well secured them on, the learner is ready for his first venture on the ice. He will probably look round for the kindly aid of some friend in his early efforts; but if he cannot find such assistance, he will have no great reason for regret It is to his advantage that he should acquire the habit of independence of all external aid, as far as possible, from the first; and if he can find some one to lead him on to the ice in the first instance, and, perhaps, to help him to Fig. 1. fall gently. This may seem strange advice, but it is quite prac- Fig. 2. but you need not trouble yourself about keeping it regular, as the ability to accomplish this will come in time. To stop yourself, you raise your toes slightly from the ice and press the heels firmly downward, being careful to throw the weight of the body forward meanwhile, or your feet will slip from under you. When you are tolerably proficient in the forward movement, you may attempt backward skating. In this you turn yourself suddenly round from the forward motion, twisting the toe of each foot inward alternately, and keeping on the inside edge of the skate. You then continue the backward progression with both feet on the ice, so that the two together describe the double curve shown in Fig. 1. The practice of this movement will be found very useful when you come to attempt figure skating. When confidence is acquired on the inside edge of the skate, you may attempt the more difficult skating on the outside edge. In this movement, when the right foot is put forward, the left shoulder must be advanced and the right arm thrown back, and vice versa, while the foot presses on the outside edge of the skate, as the name of the movement implies. The one foot should be lifted directly the other is put down, or you will find yourself turning to the inside edge. The strokes should be short at first, but as you proceed you will be able to take a stronger sweep, and describe a bolder curve, until your line of progression resembles that shown in Fig. 2. This is known as the "Dutch roll," being the method usually practised in Holland, where the inhabitants are nearly all proficient skaters. The upper curves in this figure are formed with the right foot, and the lower by the left. up. the Remember in all kinds of skating to keep your gaze directed forward in the direction you mean to follow, and never suffer it to be turned upon the ground. The head should be held body inclined in the direction in which the skater is proceeding, as already mentioned; the legs kept straight when not bent at the moment of the stroke, and the arms used moderately as a counterpoise to the legs in keeping the balance of the body. Some further instructions, with especial reference to figure skating, will be given in our next paper. NATURAL HISTORY OF COMMERCE. CHAPTER IX. (continued.) FOREIGN PRODUCE: EUROPE. The Danube flows through countries less advanced than France and England, and their condition reflects itself in their produce. The surplus for interchange consists almost wholly of raw materials. Nearly a Productions of France and Italy-Danubian Region-Holland and fourth part of Austria, and probably a larger propor Belgium-Germany. Mediterranean Seaboard. France and Italy.-France, like England, is a manufacturing nation, importing raw materials and sending out finished goods. Nevertheless, it is the chief wine country, and produces beet sugar in abundance, besides a surplus of corn, madder, and fruits for export, as well as many millions of eggs. The mulberry trees that cover a large part of the southern provinces are the basis of an important branch of national industry. Although the quantity of silk does not equal that of Italy, yet the silk fabrics of France have hitherto been unexcelled. Iron and coal, lead and zinc, are amongst the minerals of the south of France, iron being specially abundant in the Pyrenean districts. France, washed by three seas, is admirably placed for interchange, which, during the reign of Napoleon III., has more than quadrupled in value and extent. Italy is naturally a land of abundance. It is also the chief silk-producing country of Europe. It yields the best olives and olive oil, straw for plaiting-Tuscan and Leghorn plait being universally admired-sumach and bark for tanning, a fine hemp fibre, and fruits like those of Spain. A large proportion of the people, however, lack the necessaries, not to speak of the comforts of life, and those who sow and reap her bountiful harvests are often without bread. Many parts of great beauty and fertility are unhealthy, and districts once crowded are now deserted, owing to pestilential malaria. The mineral resources of Italy, though vast, are to a great extent undeveloped. They comprise marbles, alabaster, serpentines, boracic acid, sulphur, rock salt, various ores, as those of copper, iron, lead, silver, mercury, and antimony, together with mineral fuel and oils. Fisheries.-The Mediterranean sea-board, both of France and Italy, is notable for its fisheries. The delicate anchovy, preserved in salt, and the sardine, preserved in oil, are exported in large quantities. The sea enclosed by Naples, Sicily, and the islands westward, is the chief scene of the tunny fishery. This fish, sought for its oil as well as for food, represents the mackerel of the British seas, as the anchovy and sardine represent the herring tribe. Along the Barbary coast and off some parts of Italy, French and Italian dredgers engage in the so-called fishery for coral (Corallium rubrum). Only in these parts is this dense and beautiful but brittle product found in abundance, employing not only the dredgers, but the lapidaries of Marseilles, Genoa, and Naples, by whom its beauties are so developed that its value often increases to double that of gold, giving rise to interchange with Persia, China, and the most distant countries. The cutting of cameos in imitation of the ancient onyx is an industry allied to that of the coral, and carried on in the same towns. It is provided for by the prevalence of beautiful gasteropod shells, such as the wing shells used in our own country for decorative purposes. Region of the Danube.-The Alps send out eastern spurs, with glacier scenery and yawning abysses only next to the main range in grandeur. These spurs, with the still more rugged Carpathians and the Turkish Balkans, give a general mountainous aspect to this region, modified by the corn plains and grazing grounds of Hungary and the marshes of the Lower Danube. Spring clothes the meadows in green, amidst which the daffodil, narcissus, and other liliaceous bulbs grow in wild, native vigour, while sheep and cattle find a rich sustenance, and add greatly to the wealth of the land. VOL. V. tion of Turkey, are in forest, where many of the finest timber trees reach their most perfect state. The oak abounds, productive of gall nuts, of valonia for dyeing and tanning, and of a sweet acorn, flavoured like the chestnut. The forests of the Austrian empire are attendant upon inexhaustible mines, for though the coal-fields, both of the Hungarian and German provinces, are far from inconsiderable, either in point of area or produce, yet the quality of the coal is bad, and wood is much used in smelting. The mines have been worked from the time of the Romans, but never extensively, and little impression appears to have been made upon the mineral stores. Iron is found in nearly all parts of the empire, especially in Bohemia, Moravia, and the other German provinces. Some of the mountains are a pure carbonate of iron, requiring to be quarried rather than mined, and native steel of the highest excellence is found in Styria. The richest European mines of quicksilver, next to those of Spain, are met with in Idria, in the province of Carniola, Austrian Germany. An old proverb says of three Hungarian towns, in allusion to the richness of their mines, that one (Neusohl) is enclosed in walls of copper, another (Schemnitz) in walls of silver, and the third (Kremnitz) in walls of gold. Wool, silk, and metals are the chief raw substances exported by Austria, whose unwise policy in restricting commerce, with the view of keeping her produce for home consumption, has resulted in the discouragement of industry, hindrance to the increase of wealth, and the promotion of extensive smuggling. Turkey and Greece produce, besides silk, madder, figs, raisins, valonia, and olive oil, some substances more especially their own, as opium, cotton, drugs, and sponge. The fisheries of this last assume, in the Egean Sea, the place of the Italian coral fishery. Strewn over the rocky floor of the clear water where the Cyclades repose, sponge cups abound, soft, elastic, absorbent, and free from spicules of flint or lime. Those obtained from the coast of Candia (Crete) are of the finest description, and under the name of Smyrna, or Turkey, or Greek sponges, command the highest price in the market. Alpine Ridge. The climate and soil of this dividing line are unfavourable to animal and vegetable produce, and nothing economically important characterises it. The mountains, as their geological structure indicates, are deficient in metals and useful minerals. The Swiss are, nevertheless, well clothed and fed; while Italy, with which Switzerland so strongly contrasts, depends upon foreign industry for the scanty supplies of half her population. Northern Slope.-While the olive and orange flourish only in the lower latitudes of this favoured zone, the vine reappears on the northern slopes, and furnishes many of the finest wines. The climate and soil are equally favourable for the cereals and for the rearing of domestic animals. Zone of Wheat and Northern Grains. Beer and Butter Countries.-The designation of the "wine and oil countries" contrasts with that of the next higher zone, whose distinctive produce has gained for it the appellation of the Beer and Butter Countries. The two descriptive beverages are linked by the cider, common for some distance on both sides of the line of division. The production of cider, beer, and butter indicates essential differences in climate, soil, and other physical conditions, from the zone of wine and oil. The shades of change upon the face of Nature are very 110 |