5. A circular plot of ground contains one acre; what is its diameter ?

We call the attention of the reader to Examples 1 and 2 under Problem XIV. It will be there observed that when the

diameter is 1 the area is 7854, and when it is 2 the area is 31416-that is to say, double the diameter produces four times the area. This we might expect. But by further comparison it will be seen that in either case the area is equivalent to the square of the diameter multiplied by the same figures, 7854.

Hence we obtain

Rule 2: The area of a circle is equal to D2X 7854.

EXAMPLE 1.-The radius of a circle is 1; what is its area? (See Example 2 under last rule.)

E = 1, .. D = 2. And D2 X 7854 = 4 x 7854 3.1416, which corresponds with the answer obtained by Rule 1.

EXERCISE 19.

1. A circular table is 59 inches in diameter; what is its area by Rule 2?

2. Find the area of a circle whose diameter is 78 yards. PROBLEM XV.-The circumference only of a circle being given, to find its area.

Rule: Multiply the square of the circumference by '07958.* EXAMPLE.-The circumference of a circle is 3:1416; what is its area ?

3-14162 079587854, nearly.

This example proves the correctness of the rule, because the circumference being = π, D = 1, and by previous rule D2 X 7854 is in this case '7854.

EXERCISE 20.

1. The circumference being 6, what is the area of the circle + 2. The circumference of a circular plot of ground is 246 yds. 1 foot 10 inches; what is its area?

3. How many square yards are contained in a circular table whose circumference is 11 feet?

PROBLEM XVI.-To find the area of the sector of a circle. (See Definition in Geometry, Vol. I., p. 53.)

ment; if it be greater, the area is the sum.

Rule 2: From the arc of the segment subtract its sine, and multiply the remainder by half the radius.

Rule 3: Area

c+ 4

4 10

h x No 1 of the segment, or its versed sine, and c is the chord. Note. This rule is approximate only.

h, in which h is the height

EXAMPLE 1.-What is the area of the segment of a circle, the number of degrees being 27 and the length of the radius 20? By Rule 1. To find the area of the sector AC BE (Fig. 23). As 360° 27°:: 1256-6 (whole area of circle): 94-24, area of sector.

2. What is the area of a segment, the length of arc being 46-58, and the whole circumference being 156 ?

PROBLEM XVIII.-To find the area of a circular zone, ABCDEF (Fig. 24).

Draw the straight lines AE, BD; the zone is then divided into a trapezoid, A B D E, and two segments, A FE, B C D. Rule 1 Find the area of the trapezoid by rule to Problem XI., and also the area of the two segments, A FE, BC D. The sum of these areas will be the area of the

zone.

1. Find by both rules the area of a sector, the length of the radius being 5 feet, and the length of the arc 20 feet.

2. The arc of a circle contains 36° 42′ 16", and the length of the radius is 4; required the area of the sector.

3. The diameter of a circle is 578 feet, and the number of degrees in the arc is 93° 48' 30"; what is the area of the sector in acres, roods, etc.?

PROBLEM XVII.-To find the area of a segment of a circle. Rule 1: When the number of degrees in the segment is given. Find the area of the sector A C B E (Fig. 23), and then the area of the triangle A B E. If the segment is less than a This number, 07958, is produced by dividing unity by 4; i.e., -='07958.

1

4=

The fact that the area appears less than the circumference is because the latter is square measure and the former lineal.

the area of the whole circle.

G

K

H

Fig. 24.

EXERCISE 23.

1. The parallel chords of a circular zone are 5 and 6, and the diameter of the circle is 20; what is the area of the zone?

2. The radius of a circle is 14, and the lengths of the parallel chords of a zone of that circle are 22 and 28. Required the area of the zone.

PROBLEM XIX.-To find the area of a circular ring, A B C D, A'B'C' D' (Fig. 25), that is, of the space included between two concentric circles.

Rule: Find the area of the interior circle, which subtract from the area of the outer circle.

This simple problem and its rule is so self-evident as scarcely to need an example. We refer our reader to Problem XIV. for all the information requisite. We give one example for practice. EXAMPLE. The diameter of the earth's orbit being (approximately) 180,000,000 miles, and that of the earth being 7,912 miles, what part of the superficial area of the orbit is occupied by a plane passing through the diameter of the earth and bounded by its circumference? Ans. 517,574,000.

As a general definition, a solid may be regarded as a body having length, breadth, and thickness, and in this sense it of course includes liquids. It is, in fact, anything which is bounded by surfaces in all directions. The measurement of a solid is called its cubical content, and it involves three separate acts of multiplication. If the figure be a cube, its content is measured by the cube of one of the lines which connect any two adjacent angles. Thus, in Fig. 28 the contents will be represented by A B3, or A c3, or c D3. Suppose, however, that the length of AB = 1, then 13 = 1; that is, the solid content of a cube whose side is unity is also unity, the difference being between lineal and solid measure. EXAMPLE. The solid content of a cube is required whose side measures 10.

10 x 10 x 10 (or 103) = 1,000, the solid measure.
EXERCISE 26.

1. What is the solid content of a cube whose side measures 2 ft. 6in. ?

In order to find the length of the side of a cube whose solid content is known, extract the cube root of the contents.

In order to find the solid content of any other form of paral lelopipedon than a cube, multiply the length by the breadth, and that product by the height.

=

EXAMPLE 1.-Required the content of a parallelopipedon whose length, breadth, and height are respectively 12, 6, and 3. 12 x 6 72; 72 x 3 = 216. Ans. EXAMPLE 2.-What is the content of a parallelopipedon whose length is 3 ft. 2 in., its breadth 2 ft. 3 in., and its height 1 ft. 4 in. ? Ans., 9 cub. ft. 864 cub. in.

The solid content of a prism or cylinder is found by multiplying the area of the end by the length.

EXAMPLE 1.-A hollow cylinder is 12 inches in diameter inside and 12 inches high; how many cubic inches of water will it contain ?

The solid content of a regular solid ring is found by multiplying the area of the cross section of the ring by its length, the length being found by multiplying the mean diameter (that is, half the sum of the inner and outer diameters) by π.

We need scarcely observe that there are many more problems bodies-as, for instance, of a circular spindle; of a spheroid, or in connection with the measurement of the content of solid the segment of a spheroid; of a paraboloid, or the frustrum of a paraboloid; of a hyperboloid, or its segment; and so on. But our subject is directed principally to a consideration of the areas of flat surfaces, so as to apply the rules to the measurement of land. We have merely and briefly called the student's attention

to the more common forms of regular solids, and shall in our next lesson commence the subject of land-surveying.

But although these menstrous and sudden growths call for our wonder and admiration of the power of Him who can thus produce such huge structures from spores which are invisible to the

LESSONS IN BOTANY.-XLV.

SECTION CXXIV.-FUNGI (continued).

THE Clathrus cancellatus, of which we spoke before, is a most remarkable fungus. Its lower member, as exhibited in the cut (Fig. 304), is white, the upper lattice-like part a.bright coral hue. It has only been found in two places in England, being an inhabitant of the south of Europe. These two places are the Isle of Wight and Torquay; in this last-named place it has been found in two localities. Mrs. Griffiths says, in describing those first discovered: "It appeared in Mrs. Travers's garden at Torquay, in rich reddish earth, formerly a plantation. When Mrs. Travers gathered the fungus it was in a ball, and before she could bring it into the house it had burst up to its height. The scarlet part had a most vivid colour till the darker part decomposed. I was so very much annoyed by the stench that I could not take more pains with the drawing." In the autumn of 1853 other specimens were found of this curious plant in another part of Torquay.

The expansive growth of fungi, and their varied habitats, must next call for a few remarks. Some of the facts supplied as by authors on the former would be considered as scarcely credible, did they rest on less worthy evidence than that which attests them. Sowerby states, that he has placed specimens of the Phallus caninus, or "stinking morel," on his window over night, in the egg-shape, and found them next day fully grown; and another author speaks of his placing Phallus impudicus within a glass vessel, and its expanding so rapidly as to shiver the glass to pieces with an explosive detonation as loud as that of a pistol. Carpenter gives an account of a paving-stone, twenty-one inches square, and weighing eighty-five pounds, being raised an inch and a half from its station by a cluster of toadstools springing up under it; and many other facts, which attest as well the explosive power as the rapid growth of fungi, are related by different authors, one having been known to attain the size of seven feet five inches in circumference, and the weight of thirty-four pounds, in three weeks, and others the weight of twelve pounds in a few days. But none of these statements, remarkable as they are, are so wonderful as one which is made by Sir Joseph Banks of a circumstance which occurred under his own roof. He states that a friend having sent him a cask of wine, which was too new and sweet for present use, it was locked up in a cellar to mature. At the end of three years, Sir Joseph, supposing that time had now done its work, proceeded to open his cellar and inspect its contents. Little did he think how time had been employed, and little did he conceive what would be the contents of that cellar. The door refused to open, and being invincible by gentle means, he had it fairly cut away; but he was no nearer effecting an entrance than before. The cellar was found to be literally full of fungous growth, which had borne the cask aloft to the ceiling, where it stuck, upheld by fungi, the produce of the wine, which had all leaked out and formed this monstrous growth!

a few hours, an organised structure of such mag itude and such complication of arrangement, we must not let our praise and adoration stop here; for in the minuter growths, which we shall soon examine, we shall find as wondrous an exhibition of surpassing skill as in these larger products. The microscopic fungi-those which, by fastening on his crops, become the bane of the farmer, and are in God's hands a means whereby he can cut off our staple article of food, and "destroy the staff of bread," under the name of "the smut in wheat" (Puccinia graminis), or cause our bean or potato crops to perish-are among the most surprising of vegetable productions, and will hereafter engage our attention.

We have already observed that the treasures of food which it has pleased God to provide for us in the fungus tribe are, if not wholly disregarded, at least by no means duly appreciated by the English. There is, perhaps, no country richer than our own in the esculent species of fungi; they abound in our woods and pastures, they grow from the ground and under the ground; they spring abundantly out of the substance of dead trees, and are often found on waste lands and heaps of rubbish, from which no other edible produce can be procured; yet, though this is the case, and more than thirty species of esculent fungi are spontaneously brought forth in England, there are only three or four of these species that are eaten by its inhabitants, all the rest of this abundant supply being allowed to rot under the trees, or to become the prey of field-mice, toads, and slugs. Throughout the continent of Europe, on the contrary, plants of this tribe are eagerly sought after by all classes of men, and form the chief, if not the sole, diet of thousands, who would otherwise be but scantily provided with aliment. But fungi are not only the tolerated food of the poorer classes, they are also highly prized by the rich man and the epicure. In Italy and Germany immense numbers of the various species of this tribe are sold in the markets, and produce an almost incredible amount of income. In Rome, so important are the fungi as an article of commerce, that there is a public officer appointed to test the species exposed for sale, and superintend this branch of the revenue; for in that market a tax is laid on all quantities of fungi presented for sale exceeding ten pounds in weight. All fungi brought into Rome are supervised by this officer, weighed, sealed up, and all destined for that day's consumption sent to a central depôt. If, among the contents of the baskets offered, any stale, maggot-eaten, or dangerous specimens are found, they are sent under escort, and thrown into the Tiber; and another remarkable circumstance is the law, that if any specimen of the common mushroom (Agaricus campestris) is found, it also is to be thrown into the river! So says an unpublished letter of Professor Sanguinetti, "Ispettore del Funghi" at Rome. It is certainly singular that the only fungus which is freely accepted in all English kitchens, and considered as the sole common kind that is honest and trustworthy, and possessed of no murderous properties, should be the one thus protested against, whether in a state good or bad! "For forty days during the autumn, and for about half that period every spring, large quantities of funguses, picked in the immediate vicinity of Rome, from Frascati Rocca di Papa, Albano, beyond Monte Mario, towards Ostia and the neighbourhood of the sites of Veii and Gabii," are brought to Rome. "The returns of taxed mushrooms alone," says Dr. Badham, "during the last ten years, give a yearly average of between sixty and eighty thousand pounds weight; and if we double this amount, which we may safely do in order to include such smaller untaxed supplies as are disposed of in bribes, fees, and presents, and reckon the whole at the rate of six baiocchi, or threepence per pound (a fair average), this will make the commercial value of fresh funguses very apparent, showing it here to be little less than £2,000 a-year." Besides this, we must consider the dried, pickled, and preserved supplies, which sell at a much higher price than the fresh, from one shilling to one shilling and threepence per pound, and also recollect that this calculation includes only the Roman market, and that every other market-place in the Italian states has its proportionate sale of this wide-spreading branch of the vegetable produce of the land.

With the above statements fully in our mind, and after

having been habitually in communication with many of the families from amongst our peasantry who were but scantily provided with daily food, we found ourselves one day in an extensive pine-wood, near Budleigh Salterton, in South Devon, and saw the ground, which was densely carpeted with the accumulated

poor from the possibility of availing themselves of this rich supply of whole. some aliment, which the bounty of God had provided for them, if they would but be persuaded to use it? It is true, that among this extensive collection of fungi there were some species which would have proved poisonous, and others which would have been but disagreeable food; yet the greater number of them were such as, if properly cooked, would have furnished, not only wholesome, but also savoury and pleasant food; and it seemed to us a great pity that they should be so wholly neglected, and left a prey to reptiles and field-mice.

some of them possess when cooked, we find the following fact stated that, "like animals, they absorb a large quantity of oxygen, and disengage in return from their surface a large quantity of carbonic acid: all, however, do not exhale carbonie acid, but in lieu of it some give out hydrogen, and others azotic

gas. They yield, moreover, to chemical analysis the several components of which animal structures are made up; many of them, in addition to sugar, gum, resin, a peculiar acid called fungic acid, and a variety of salts, furnish considerable quantities of albu. men, adiposine, and osmazome, which last is that principle that gives its peculiar filavour to meat gravy." Fungi are considered to be highly nutritious, and are said by many of the faculty to be easy of digestion. This latter opinion, though strongly supported by many foreign medical men, is certainly quite in opposition to the generally received opinion on that subject in England, and also to the ideas of ancient writers. Gerard, the quaint old herbalist. says: "Some mushroomes grow forth of the earth: others upon the bodies of old trees, which differ altogether in kinds. Many wantons that dwell near the sea, and have fish at will, are verie desirous for change of diet, to feed upon the birds of the mountaines; and such as dwell upon the hill or champion grounds do long after sea fish; many that have plenty of both do hunger after the earthie excrescences called mushroomes; whereof some are very venomous and full of poison, others not so noisome, and neither of them very wholesome meat." And again, "Galen affirms that they are all very cold and moist, and therefore do approach unto a venomous and murtherous facultie, and ingender a clammy, pituitous, and cold nutriment, if they be eaten. To conclude, few of them are good to be eaten, and most of them do suffocate and strangle the eater. Therefore I give my advice to those that love such strange and new-fangled meates, to beware of licking honey among thornes, lest the sweetnesse of the one do not countervaile the sharpnesse and pricking of the other."

The chemical structure of fungi is said to be the most highly animalised, or, in other words, to partake more of the nature of animal composition than that of any other vegetable. Besides the intimations of this circumstance that are afforded by the smell of some of the species in decay, which partakes much of the character of that of putrid meat, and the strong meat-like flavour which

NATURAL HISTORY OF COMMERCE.

CHAPTER XII.

NATURE AND MAN AS AGENTS OF CHANGE.

We have now traced the relation throughout the earth, (1) between geological conditions and mineral produce; (2) between climate and soil on the one hand, and organic produce on the other. We have seen how contour, vertical relief, and other physical facts modify climate and soil, and consequently animal and vegetable produce. We have also been led to observe that a study of the geology of any region is auxiliary to a knowledge of its flora and fauna.

The great mountain ridge of the New World presents no barrier between the equator and the poles, as the Himalayas and their adjuncts offer in the Old World. The dispersion of plants and animals is therefore limited by more elastic conditions, and the separation of zones is marked by less decided lines. The tropical waters of the Caribbean Sea divide America into a northern and southern continent, closely corresponding; each division being related to the other by bands of analogous climate and produce, which are but portions of bands similarly crossing the Old World, and encircling the

earth!

There is abundant geological evidence to prove that the zones of the earth have been subject to change both in climate and produce. The agents of change have been Nature and man. The great plains of America are geologically recent. Coal-beds are found in the Arctic regions, the fossil flora of which, composed of a preponderance of conifera, indicates a climate corresponding to that of mid-Europe, and proving that in the long cycles of the earth's physical history and its successive oscillations, the poles whose frosts we often call eternal, have aforetime and more than once borne excess of heat. The vibrations of the earth in reference to its axis, slow though they be, are persistent; and although intervals of thousands of years are required to make perceptible comparisons, yet the equilibrium of our planet goes through a long period of unresting change. At the present time the Arctic coasts are rising and the bed of the Pacific is sinking, in obedience, it is believed, to the law requiring the centre of gravity of the earth's mass to be maintained by the mobility of the ocean. The nature of the changes thus brought about may be illustrated by a supposition easy to comprehend. If the relation between America and the Atlantic were gradually to alter so that the sea-level rose 300 feet, the llanos of the Orinoco would be covered. If it were 1,100 the sea would wash the base of the Andes, and only leave those mountains and the highland of Venezuela, the Guianas, and Brazil above the waves.

Man finds work ever ready to his hand. By diligent labour, guided by intelligence, he can modify the minor aspects of Nature, and obtain from her bounty an indefinite increase of enjoyment. He cannot alter the past or arrest the future, but he may shape the issues of both to his advantage. What are the limits of our power, and how may we best use it to promote wellbeing? Such are the inquiries which the course of study we have pursued should aid us to answer.

Nothing more beautifully shows the harmony of natural laws than the modifications of the forms of life by the change of conditions. We fell a forest, and the timid browsers lose their shelter and food and disappear; the wild beast is deprived of its covert and prey, and is seen no more; even so birds migrate to districts where insects and berries abound. We cultivate a plain, and the

VOL V.

grub of the cockchafer begins its havoc with the corn roots, and the earthworm its system of under-tillage, till, attracted by their prevalence, the familiar forms of our common birds are seen, and the balance of vegetable and animal life is restored. The sparrow was unknown in Russia last century; but the rapid progress of corn this bird to spread over the empire, even as far as Siberia. Partridges, again, whose food is found in the corn-fields of England and France, have recognised the high husbandry of Scotland, and are met with at Inverness, the limit of British wheat growth. Food, therefore, is seen to be the link between the flora and fauna of the climatic zones.

Many illustrations might be submitted of the effects of human agency in modifying the aspects of nature, sometimes intentionally arranged, sometimes otherwise. Mr. Grierson, at the meeting of the British Association in 1866, read a paper referring to the destruction of plantations at Drumlanrig, in Dumfriesshire, by the voles, commonly called rats, which are the pest of Sweden. They appear to be migratory in their habits, and occasionally increase in myriads. From the recent slaughter of rapacious birds, such as owls, hawks, and eagles, which Nature has appointed to bound the` unlimited fecundity of the rodentia, the voles found a safe field for action. They principally destroyed the young oak and ash, gnawing a ring of bark near the roots and beneath the grass; the trees being unable to resist such attacks until after at least twelve years' growth.

By our acquaintance with the facies of a floral region we are able to judge when and where we can with profit introduce or transfer the plants of one country to another, even to another hemisphere. Thus it is that we have spread the useful food plants, fruits, fibres, and timbers, or strewn our colonies with wild flowers, associated with the thoughts of home.

The vegetable kingdom is full of striking examples. The fruits of Europe, mostly of Asiatic origin, were removed westward in the same zone, and subsequently to the New World. The diffusion is carried still further into the zones of the southern hemisphere. The vine now flourishes in South Africa and Australia. Grains, either tropical or northern, have gone with man into every habitable clime. Maize has enlarged its area in the three continents of the East, and rice has spread almost as widely in the West. We owe to Chili the potato, which has lightened existence to extra millions of mankind. The pine-apple was a native of the Bahamas and Bermudas, but is now plentiful round the Mediterranean. Tobacco, unknown till the sixteenth century, belts both sides of the equator far beyond the tropics. Cloves and pepper are acclimated in each of the Indies, though native only to the East. Coffee, also an indigene of Arabia or Abyssinia, has sped throughout the tropical zone. Thus, also, with the fauna, we have aided Nature in the distribution of her productions, enriching each zone with the representative species of its corresponding zone. The wild horses and cattle of South America seem destined to exterminate the native llama. English sheep in Australia have driven the kangaroo inland, and threaten its extinction. The effect of introducing our domestic animals into other countries has been to increase our resources for food and clothing, to add to wealth, and to the duration of human life.

The

Nature has arranged the climatic zones in a manner whose simplicity and unity of working fill our minds with an exalted pleasure. Oceans come between the continents, and obstruct the passage of certain forms of life; yet many thousands of miles distant the conditions of being are only modified, and we meet, not the same species, but representative ones, whether of animals or plants. The animals of the Old World, both

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