other rose. To this raised region was given the name of Ullah Bund, "The Mount of God." Through this Ullah Bund the Indus had to force a cutting. The cutting revealed the fact that the whole bed of the soil consisted of clay with marine shells, proving that at a previous period the region had been a bed of the sca. The most remarkable instance of the repeated processes of elevation and subsidence in the same district, is found in the Bay of Baiæ, to the north of Naples. In that bay is situated the town of Puzzuoli, formerly called Puteoli. It is the place where Paul landed after his voyage on his visit to Rome. And after one day the south-wind blew, and we came the next day to Puteoli, where we found brethren, and were desired to tarry with them seven days." (Acts xxviii. 13, 14.) Were he to land there now, he would not know the district; for, in the north of the bay, an entire mountain, called Monte Novo, has been raised up, which was not there at the time of his visit. much studied by scientific geologists, and it is now ascertained that, for the last thirty or forty years, a gradual sinking of the coast is again going on, and that the floor of the temple becomes frequently covered again by water from the sea. LESSONS IN ARITHMETIC.-No. XVI. CONTRACTIONS IN DIVISION. THE operations in division, as well as those in multiplication, may often be shortened by a careful attention to the application of the preceding principles. CASE I. When the divisor is a composite number. EXAMPLE 1.-A man divided 837 crowns equally among 27 persons, who belonged to 3 families, each family containing 9 persons: how many crowns did cach person receive Explanation. Analysis.-Since 27 persons received 837 crowns, each one must At Puzzuoli, close by the sea-shore, are the remains of a have received as many crowns as the number of times that 27 is conmagnificent building-whether of a bath or a temple has not tained in 837. But as 27 (the number of persons), is a compobeen decided, but it is known all over the world as the Templeite number whose factors are 3 (the number of families), and 9 of Serápis. The building was quadrangular, 70 feet in di- (the number of persons in each family), it is obvious we may first ameter. The roof was supported by 46 pillars, 24 of granite, find how many crowns each family received, and then how many and 22 of marble, each consisting of a single block. Of these each person received. 22 marble columns, three remain standing, the tallest of them being 42 feet high. The surface of these columns is smooth and uninjured up to about twelve feet from the pedestal. Then begins a series of perforations and holes in the marble. These holes and perforations continue upward in a regular band round the column to the height of nine feet, and then cease; and the surface continues smooth all the way to the summit. The upper edge of the perforated band is now 23 feet above the level of the sea. How came these perforations into the columns? All the holes are deep, and in the shape of a pear,-i. e., very narrow at the entrance, but become larger as it enters the marble. It is evidently the work of a species of mussels called modiola lithophaga-marine shell-fish which eat into stones. A large number of these holes contain to this day the shells of the fish which perforated them, though many have been emptied by travellers. Divisor { Ans. Operation. If 3 families received 837 crowns, 1 family must have received as many crowns as 3 is contained times in 837; but 3 is in 837, 279 times. That is, each family received 279 crowns. Again, if 9 persons (the number in each family) received 279 crowns, 1 person must have received as many crowns as 9 is contained times in 279; and 9 is in 279, 31 times. Therefore 31 crowns is the share of each person. To divide by a composite number. Rule: Divide the dividend by one of the factors of the divisor, then divide the quotient thus obtained by another factor; and so on till all the factors are employed. The last quotient will be the answer required. To find the full remainder. Rule: If the divisor is resolved into but two factors, multiply the last remainder by the first divisor, and to the product add the first remainder, if any; the sum will be the compound remainder. mainder by all the preceding divisors, and to the sum of their When more than two factors are employed, multiply each reproducts add the first remainder; the sum will be the fan eThe full remainder may also be found by multiplying the quotient by the divisor, and subtracting the product from the mainder. How did the mussels come to attack these columns? and ow did they come to limit their operations just to a band nine eet in width? There can be no doubt that the temple to which they belonged, and the ground on which they are placed, were submerged under the water of the sea. When they were in this sunken state, the basement was protected from the boring mussels by masses of rubbish, tufa, and silt which the sea-water washed around them, and the upper part of the columns was beyond the reach of the sea. The perforations in this column prove-1, that this coast has, since the temple This contraction is exactly the reverse of that in multiplication. was built, sunk beneath the level of the sea; 2, that the same The quotient will evidently be the same, in whatever order the coast has been again elevated; 3, that the movement down-factors are taken. dividend. ward and again upward was more than twenty feet; and 4, EXAMPLE 1-A man bought a quantity of clover seed amount. that the elevation and subsidence was so gradual as to permit ing to 507 pints, which he wished to divide into parcels containing these columns to maintain their erect position. 64 pints each: how many parcels can he make? Since 64 2X8X4, we divide by the factors respectively. Operation. 2)507 Dividend. 8)253-1 rem. 4)31-5 rem. 64 Quotient Now, all these changes of this temple have transpired since the time that Paul landed at Puteoli. Among the ruins, inscriptions have been discovered, which record that certain embellishments of marble were conferred on the building by Divisor Septimius Severus and Marcus Aurelius. The Emperor Severus died A. D. 211. This proves that this temple was in its original position at the commencement of the third century of our era. In A. D. 1198, in consequence of an eruption of Solfatara, in that neighbourhood, a subsidence of the coast took place, and the temple sank with it, and the columns came within the reach of the boring mussels. They continued for some time to sink lower and lower, and as they sank the mussels carried on their perforations higher and higher. They must have continued in this submerged state till near the middle of the sixteenth century, for in 1530 it is well known that the whole of that coast was covered by the sea. In 1538 an earthquake, connected with Vesuvius, agitated this district, threw up in one night on this shore a mountain 450 feet high, raised the coast on which the temple is built to the height of 20 feet, and formed a new tract of coast six hundred feet in breadth. It was then that these columns were raised beyond the reach of The mussels of the sea. These columns have been latterly First remainder = 1 pt. 5X2 = 10 pts. Full remainder 59 pts. Quotient. 7 parcels, and 59 pints over. Dividing 507, the number of pints, by 2, gives 253 for the quotient, or distributes the seed into 253 equal parcels, leaving I pint over. Now the units of this quotient are evidently of a different vain. from those of the given dividend; for since there are but half as many parcels as at first, it is plain that each parcel must contain 2 pints, or 1 quart; that is, every unit of the first quotient contains two of the units of the given dividend; consequently, every unit of it, as 5, that remains will contain the same; therefore this remainder must be multiplied by 2, in order to find the units of the given dividend which it contains. Dividing the quotient 253 parcels, by 8, will distribute them into 31 other equal parcels, each of which will evidently contain 8 times the quantity of the preceding, viz: 8 times 1 quart 8 quarts, or 1 peck; that is, every unit of the second quotient contains 8 of the units in the first quotient, or 8 times 2 of the units in the given dividend; therefore what remains of it, as 3, must be multiplied by 8×2, or 16, to find the units of the given dividend which it contains. In like manner, it may be shown, generally, that the division by each successive factor reduces each quotient to a class of units of a higher value than the preceding; that every unit which remains of any quotient, is of the same value as that quotient, and must therefore be multiplied by all the preceding divisors, in order to find the units of the given dividend which it contains. Finally, the several remainders being reduced to the same units as those of the given dividend according to the rule, their sum must evidently be the compound remainder. EXERCISES. 1. How many acres of land, at 35 crowns an acre, can you buy for 4650 crowns? 2. Divide 16128 by 24. 3. Divide 17220 by 84. 4. Divide 25760 by 56. 5. Divide 91080 by 72. 6. Divide 142857 by 112. 7. Divide 123456 by 168. CASE II.-When the divisor is 1 with ciphers annexed to it. It has been shown that annexing a cipher to a number increases its value ten times, or multiplies it by 10. Reversing this process-that is, removing a cipher from the right hand of a numberwill evidently diminish its value ten times, or divide it by 10; for, each figure in the number is thus restored to its original place, and consequently to its original value. Thus, annexing a cipher to 15, it becomes 150, which is the same as 15X10. On the other hand, removing the cipher from 150, it becomes 15, which is the same as 150-10. In the same manner it may be shown, that removing two ciphers from the right of a number, divides it by 100; removing three, divides it by 1000; removing four, divides it by 10000, &c. Hence, To divide by the numbers 10, 100, 1000, &c. Rule: Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor. The remaining figures of the dividend will be the quotient, and those cut off will be the remainder. EXERCISES. 1. In one pound there are 10 florins; how many pounds are there in 200 florins? In 340 florins? In 560 florins? 2. In one metre there are 100 centimetres; how many metres In 765000 centimetres? In are there in 65000 centimetres ? 4320000 centimetres? 3. Divide 26750000 by 100000. 4. Divide 144360791 by 1000000. 5. Divide 582367180309 by 100000000. CASE III.-When the divisor has ciphers on the right hand. EXAMPLE.-How many hogsheads of molasses, at 30 crowns apiece, can you buy for 9643 crowns? The divisor 30 is a composite number, the factors of which are 3 and 10. We may, therefore, divide first by one factor and the quotient thence arising by the other. Now cutting off the righthand figure of the dividend divides it by ten, consequently dividing the remaining figures of the dividend by 3, the other factor of the divisor, will give the quotient. Operation. Divisor 30)964|3 Dividend Quotient 321 * Explanation. We first cut off the cipher on the right of the divisor, and also cut off the right-hand figure of the dividend; then dividing 964 by 3, we have 1 remainder. Now, as the 3 cut off is part of the remainder, we therefore annex it to the 1. Ans. 321 hogsheads. When there are ciphers on the right hand of the divisor. Rule : Cut off the ciphers, also cut off as many figures from the right of the dividend. Then divide the other figures of the dividend by the remaining figures of the divisor, and annex the figures cut off from the dividend to the remainder. true remainder. Explanation. We first double the dividend, and then divide the product by 10, which is done by cutting off the right-hand figure. But since we multiplied the dividend by 2, it is plain that the 6 cut of is 2 times too large for the remainder; we therefore divide it by 2, and we have 3 for the When the divisor is the number 5. Rule: Multiply the dividend by 2, and divide the product by 10. When the figure cut off is a significant figure, it must be divided by 2 for the true remainder. This contraction depends upon the principle that any given divisor is contained in any given dividend, just as many times as twice that divisor is contained in twice that dividend, three times that divisor in three times that dividend, &c. 1. Divide 6035 by 5. 3. Divide 8450 by 5. 2. Divide 32561 by 5. 4. Divide 43270 by 5. CASE V.-When the divisor terminates in 5. Rule: To divide by 15, 35, 45, 55, &c. Double the dividend, and divide the product by 30, 70, 90, 110, &c., as the case may be. This method is simply doubling both the divisor and dividend, We must therefore divide the remainder, if any, by 2, for the true remainder. 1. Divide 1256 by 15. 2. Divide 3507 by 45. 3. Divide 2673 by 35. 4. Divide 7853 by 55. CASE VI.-When the divisor terminates in 25 or 75. To divide by the number 25. Rule: Multiply the dividend by 4, and divide the product by 100. This process is obviously the same as multiplying both the dividend and divisor by 4. Hence, we must divide the remainder, if any thus found, by 4, for the true remainder. 1. Divide 2350 by 25. 3. Divide 4860 by 25. 4. Divide 94880 by 25. Rule: 2. Divide 42340 by 25. To divide by the number 125. Multiply the dividend by 8, and divide the product by 1000. This contraction is multiplying both the dividend and divisor by 8. For the true remainder, therefore, we must divide the remainder, if any, by 8. 1. Divide 8375 by 125. 2. Divide 25426 by 125. To divide by 75, 175, 225, or 275. Rule: Multiply the dividend by 4, and divide the product by 300, 700, 900, or 1100, as the case may be. This contraction is multiplying both-divisor and dividend by 4. For the true remainder, divide the remainder, if any thus found, by 4. 1. Divide 1125 by 75. 3. Divide 2876 by 175. 4. Divide 8250 by 275. 2. Divide 3825 by 225. The preceding are among the most frequent and useful modes of contracting operations in division. Various other methods might be added, but they will naturally suggest themselves to the inventive student, as opportunities occur for their application. EXERCISES. 1. How long would it take a vessel sailing 100 miles per day to circumnavigate the earth, whose circumference is 25000 miles? 2. The distance of the Earth from the Sun is 95,000,000 of miles: how long would it take a balloon, going at the rate of 100,000 miles a year to reach the sun? 3. Divide 467000000000 by 25000000000. THE POPULAR EDUCATOR. 24. 7825600-80000. GENERAL PRINCIPLES IN DIVISION. From the nature of division, it is evident, that the value of the quotient depends both on the divisor and the dividend. I. If a given divisor is contained in a given dividend a certain num. ber of times, the same divisor will obviously be contained, in double that dividend, twice as many times; in three times that dividend, tre as many times, &c. Hence, If the divisor remains the same, to multiply the dividend by any number, is in effect multiplying the quotient by that number. Thus, 6 is contained in 12, 2 times; in 2 times 12 or 24, 6 is contained 4 times (i. e. twice 2 times); in 3 times 12 or 36, 6 is contained 6 times (i. e. thrice 2 times); &c. II. Again, if a given divisor is contained in a given dividend a certain number of times, the same divisor is contained in half that dividend, half as many times; in a third of that dividend, a third as many times, &c. Hence, If the divisor remains the same, dividing the dividend by any number, is in effect dividing the quotient by that number. Thus, 8 is contained in 48, 6 times; in 48-2 or 24 (half of 48), 8 is contained 3 times (i. e. half of 6 times); in 48-3 or 16 (a third of 48), 8 is contained 2 times (i. e, a third of 6 times); &c. III. If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend, twice that divisor is contained only half as many times; three times that divisor, a third as many times, &c. Hence, If the dividend remains the same, multiplying the divisor by any number, is in effect dividing the quotient by that number. Thus, 4 is contained in 24, 6 times; 2 times 4 or 8 is contained in 24, 3 times (i. e. half of 6 times); 3 times 4 or 12 is contained in 24, 2 times (i. e. a third of 6 times); &c. IV. If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend, half that divisor is contained twice as many times; a third of that divisor, three times as many times, &c. Hence, If the dividend remains the same, dividing the divisor by any number, is in effect multiplying the quotient by that number. Thus, 6 is contained in 36, 6 times; 6-2 or 3 (half of 6), is contained in 36, 12 times (i. e. twice 6 times); 6-3 or 2 (a third of 6), is contained in 36, 18 times (i. e. thrice 6 times); &c. V. From the preceding articles, it is evident that any given divisor is contained in any given dividend, just as many times as twice that divisor is contained in twice that dividend; three times that divisor in three times that dividend, &c. Conversely, any given divisor is contained in any given dividend just as many times, as half that divisor is contained in half that dividend; a third of that divisor, in a third of that dividend, &c. Hence, If the divisor and dividend are both multiplied, or both divided by the same number, the quotient will not be altered. Thus, 6 is contained in 12, 2 times; 2 times 6 is contained in times 12, 2 times; 3 times 6 is contained in 3 times 12, 2 times, &c. Again, 12 is contained in 48, 4 times; 12-2 is contained in 48 2,4 times; 12-3 is contained in 48-3, times, &c. VI. If the sum of two or more numbers is divided by any number the quotient will be equal to the sum of the quotients which will arise from dividing the given numbers separately. Thus, the sum of 12+18=30; and 30-6-5. Now, 12-6=2; and 18÷÷6-3; but the sum of 2+3=5. Again, the sum of 32+24+40-96; and 96-8-12. Now, 32-8=4; 24-8=3; and 40÷85; but 4+3+5=12. CANCELLING FACTORS. DEFINITION.--The method of contracting arithmetical operatims, by rejecting equal factors, is called CANCELLING FACTORS. We have seen that division is finding a quotient, which, multi plied into the divisor, will produce the dividend. If, therefore, the dividend is resolved into two such factors that one of them is the divisor, the other factor will, of course, be the quotient. Suppose, for example, 42 is to be divided by 6. Now the factors of 42 are 6 and 7, the first of which being the divisor, the other must be the quotient. Therefore, cancelling a factor of any number divides the number by that factor. CASE 1.-When the dividend is the product of two factors, one of which is the same as the divisor. Rule: Cancel the factor common to the dividend and divisor; the other factor of the dividend will be the answer. NOTE. The term cancel signifies to erase or reject. Common Method. Divisor 34)952 (28 Quotient 68 272 tient, as by the common method. By Case I. 34x28 34 28 Quotient Cancelling the factor 34, which common both to the divisor and dividend, we have 28 for the que This process may be applied with great advantage to that class of examples and problems, which involve both multiplication and division, that is, which require the product of two or more numbers to be divided by another number, or by the product of two or more numbers. 1. Divide 3420 or 76×45 by 76. 3. Divide 5103 or 63×81 by 81. 2. Divide 5330 or 65 X82 by 82. 4. Divide 6395 or 95X73 by 95. EXAMPLE 2.-Divide the product of 3780 or 45 times 84 by 9. composed of the factors 84X5X9. We may therefore cancel 9, Analysis.-The factor 45=5X9; hence the dividend 3780 is which is common both to the divisor and dividend, and 84 × 5, the other factors of the dividend, will be the answer required. Operation. 84×5×9 -=84×5=420. Proof. 3780 9 =420. 7. Divide 810 or the product of 45 × 18 by 90 or 18X5. 45X18 Operation. 18x55-9. Proof. 810 for cancelling the same or equal factors in the divisor and dividend, Rule: those remaining in the dividend by the product of those remaining 8. Divide 15X7X12 by 5×3×7X2. 10. Divide 75x15x24 by 25×3×6×4×5. The four preceding rules,-viz., Addition, Subtraction, MultipliRULES of Arithmetic, because they are the foundation or basis of cation, and Division are usually called the FoUR FUNDAMENTAL all arithmetical calculations. LESSONS IN NATURAL HISTORY.-No. IX. THE BEAR. THE bear belongs to a section of the animal kingdom to which has [Order CARNIVORA, genus URSUS.] been given the name of Plantigrades, from their applying the enraising themselves on their hinder limbs, and maintaining with ease tire sole of the foot to the ground, so as to have the free power of an upright position; their motions are slow and heavy; their habits are generally nocturnal; and in the northern regions, they usually pass the winter in a lethargic state, concealed in holes in the ground. Some of the species are able to use their fore-feet in conveying food to their mouths, or in seizing hold of objects. Their claws are strong, blunt, and well adapted to digging and climbing. able to ascend trees in search of prey, and also to escape the purIn consequence of their adaptation to the latter purpose, bears are don; and then they shared the popularity of Punch with gaping crowds, as will be still remembered by the elders of the present generation. The signs of the "Black Bear," in Piccadilly, "Taylor's Bear," in Whitechapel, the "White Bear," and the "Bear and Ragged Staff," are relics of that time. Stories, also, were afloat of bears being made to stand upon hot iron, and to undergo 381 female brings forth her young, and rears them during this period. It is said that there is nothing found in the stomach or bowels of the bear when he is tracked to his winter lair and killed; that he eats nothing for some days before retiring, and that then he is quite fat; but when he reappears in spring, he is very meagre and exhausted. a precipice, the bear always moves backwards; thus resembling a human being in his cautious descent. He is a good and rapid swimmer, and is fond of bathing during the heat of summer. A Savoyard, who obtained his living by exhibiting a monkey and a bear, gained so much applause from his tricks with the monkey, that he was encouraged to practise some on the bear; but being dreadfully lacerated in the attempt, and rescued with great difficulty from the grasp of Bruin, he exclaimed, "What a fool was I, not to distinguish between a monkey and a bear! A bear, my friends, is a very grave kind of personage, and, as you plainly see, does not understand a joke!" The white or Polar bear is a much larger animal than any of the species found on land. One measured by Captain Lyon, in his northern expedition, was found to be eight feet seven inches and a half in length, and it weighed, no less than sixteen hundred pounds. Describing this animal in the frozen regions of the North, Thomson says,— "There through the piny forest half-absorpt, Rough tenant of these shades, the shapeless bear, This creature seems especially fitted to spend its existence in a liquid element. It is much larger in the body than other bears, and its legs are shorter and thicker. It also differs much in the shape of the head; when viewed in profile, the line is nearly raight, and the upper part of the cranium is rather depressed. The muzzle is broader and thicker than in other species. The neck is nearly equal in thickness to the head. The fur is very long and thick, of a bright white beneath, and of a yellowish tinge on its upper surface. The sides of the feet are thickly covered with hair, which gives the animal a firm footing on the ice; and the claws are short, black, and nearly straight. On Berund Island, situated between Spitzbergen and the North Cape, white bears are numerous. Vessels go from Norway to this island almost every year, for the purpose of killing these animals during the winter. "Two men, with lances," says Mr. Laing, "can always despatch the animal without difficulty, one taking him in front, and the other on the side. They sometimes use small dogs, as in attacking the common bear in Norway. Among the whole of the ursine tribe the most tender parts are those behind, exposed when they walk, and vulnerable even by the smaller animals. When the dogs bark and attack the bear behind, he sits down instinctively to cover his hinder parts, and to defend himself with his fore-paws." That the maternal feeling of the bear is strong is evident from the following instance. When the Carcase frigate was locked some years ago in the northern ice, a she-bear and her two cubs, nearly as large as herself, came towards the vessel. The crew threw to them great lumps of sea-horse blubber. The old bear fetched these away singly, and divided them between her young ones, reserving only a small piece for herself. The sailors shot the cubs as she was carrying the last portion, and wounded herself. She could just crawl with it to them, tear it in pieces, and lay it before them. When she saw they did not eat, she laid her paws first on one, and then on the other, and tried to raise them up, moaning piteously all the while. She then moved from them, looked back, and moaned, as if for them to follow her. Finding they did not, she returned, smelt them, and licked their wounds; again she left them, and again returned, and then went round them pawing and moaning, with signs of inexpressible fondness. At last, she raised her head towards the ship, and uttered a growl of despair, when a volley of musket-balls terminated her life. The bears of Kamtschatka live chiefly on fish, which they procure for themselves from the rivers. A few years ago, the fish became very scarce. Emboldened by famine and hunger, the bears, instead of retiring to their dens, wandered about, and sometimes entered the villages. On a certain occasion one of them found the outer door of a house open, and entered it, the gate accidentally closing after him. The woman of the house had just placed a kettle of boiling water in the court. Bruin smelt it, but burnt his nose. Provoked at the pain, he vented all his fury on the tea-kettle. He folded his arms round it, pressed it with his whole strength against his breast in order to crush it, but this, of course, only burnt him the more. The horrible growling which the rage and pain forced on the poor animal now brought the neighbours to the spot, and Bruin, by a few shots, was put out of his misery. To this day, however, when anyone injures himself by his own violence, the people of the village call him "the bear and the tea-kettle." In addition to the species we have described there are others in different countries. America has three species of bear: the black, the cinnamon, and the grizzly. The black bear is smaller than the European species, and his glossy fur is much prized as an article of commerce. The grizzly bear is the most formidable and ferocious of the tribe; he is a native of the state of Missouri and the Rocky Mountains. The cinnamon is considered by some naturalists a variety of the black bear. India has several species of bear, as the Malay, Tibet, &c. They agree with the rest of the tribe in their general habits, but are of less size, and have short close fur. Their claws are remarkably long and curved, and as the form of the body is light, they are able to climb with great facility. In the mountains of that country there is also the sloth bear. This creature has the singular power of protruding or contracting the lips, which have great mobility, and are employed to reach or collect its food, consisting, it is said, of white ants, honey, and vegetables. Having claws which are long and powerful, it can not only readily dig up some of its means of subsistence, but it can also excavate holes and dens as places of its retirement. LESSONS IN GERMAN.-No. XIII. SECTION XXV. Dürfe u expresses a possibility dependent upon the will of another, or upon a law. Ex.: Ich darf diese Blumen nicht pflücken; I cannot (I am not allowed, permitted to) pluck these flowers. Der Bauer tarf nicht fischen; the peasant is not allowed (by law) to fish. Sch tarf diese Fruchte essen, aber ich kann ste nicht erreichen; I can (have the right to) eat these fruits, but I cannot obtain (get at) them. (§ 83. 1. 2.) CONJUGATION OF THE PRESENT AND IMPERFECT OF dürfen. Present. II Mogen expresses a possibility dependent on the will of the subject or the speaker. Ex.: Er mag geben; he can (may, is at liberty to) go Sie mögen gehen; you may (have permission to) go. Ich mag ihn nicht sehen; I do not wish to see him. Das mag ich nicht glauben; I do not like to believe that. (§ 83. 4.) III. Mogen, like “may,” denotes a concession on the part of the speaker. Ex.: Gr mag ein treuer Freund sein; he may be a true friend. Sie mögen es gethan haben; they may have done it $ 83 4.) |