sermons LESSONS IN GEOLOGY-No. I. HOW TO BECOME A GEOLOGIST. Turks are two methods of becoming a Geologist. The one pebbles that you see in the gravel ever larger than they are is by observing geological facts, and the other by reading now? How came they to be so small? and so round? If you geological works. Both of these methods must be adopted travel along a valley, how is it that the pebbles in the upper and combined, if you wish to succeed in the study of geology. part of it are large-and that those in the lower part of it be come gradually less and less as you approach the sea, till at si. By Observation. The first method of studying geology is personal observation. last they are mere sand or mud ? Remember that for all these There is no science that teaches you to make use of your eyes furnishes them. things there are reasons, and that the science of geology more completely and habitually than geology. The " It is possible that sometimes, in the summer, you make a . in stones" are never to be heard, but they are always to be long excursion by one of the railways. On such a journey, read, and read with your own eyes. This personal observa- you not only pass over a great variety of superficial soil, but tion is to be directed to the different geological materials around you travel through deep cuttings in different rocks; such cutyour own neighbourhood, and to the geological character of tings, as geologists would call fine " sections for studying gco logy.” If, on these excursions, you make proper use of your any district or country, across which you may be travelling. eyes, you will learn much of the alphabet of geology; which, Do you wish to become a geologist? If you do, as soon as in the course of a short time, you will, by attention and perseyou shall have read this article, take up the very first stone, or verance, be able to put together, in such a manner, into syl. fragment of a stone, that you can pick up in your way. As lables, and words, and sentences, as will utter to you the great and delightful truths of science. you examine the stone, ask yourself a few questions about it. Let me suppose that you live in London, and that, upon some If from these questions you at first learn nothing but your own holiday in the week, you make an excursion by railway to ignorance, you must not be discouraged; for that discovery is Brighton. I mention this, as it is the most common excursion the best means for creating and whetting a keen appetite for made by artisans and others. As you travel along, you can geological knowledge. mark the different rocks through which you pass, without, for Look at the stone again, and ask : What is it? What is the that smile on each side of you. a moment, losing the enjoyment of the charming landscapes name of it? Where did it come from? How did it come From the London-bridge station to New Cross, you ride here over the dark-looking mould which the gardeners find so well Perhaps it is piece of chalk. What is chalk? How adapted to the growth of vegetables. As soon as you pass does chalk appear under the microscope ? Is all chalk white under the bridge at New Cross, you enter a very deep cutting In what English counties is chalk found ?-Or the fragment in clay is this called ? How is it that, if you took a walk to in a high bank of clay. How is this? What is clay? What your hand may be a flint. What is flint ? Was it once soft? HAMPSTEAD by Haverstock-hill, or made a short start by the In what kind of rock is it originally found? If it is round, Great Northern Railway, you would come to the same clay? what rounded it as if it were water worn? If water worn, Was the clay at New Cross, and that at HAVERSTOCK-NILI, when and where could water have acted upon it? Do flints clays that once lay between the two places ? Has the Thames, ever one continuous bed? If so, what has become of all the ever contain fossils? How came they there?- Your pebble or any other water, scooped it out and carried it away! may be a sandstone ? What is sand? What is the difference Near CROYDON, you come to beds of gravel. How did that between sand and clay. What has given the colour to the gravel come there? What gives the tint of olive green to all sandstone? Is it soft or hard? How many kinds of sandstone that gravel? What has made those deep beds of clay, through are there :-It may be that the piece you have picked up is a which you have passed, now cease altogether? When you pass limestone. How came it to be produced? What districts in the Stoat's Nest you come ugain into deep cuttings, not inlay, as before, but in chalk. In the upper part of the cutting you England abound with rocks of limestone? How is it that see a black line continuing on both sides for miles. What is these rocks are always in layers or beds? How came shells that line? It is a layer of Aint looking as regular as a line of to be imbedded in them? How is it that some limestone is dark-coloured brick placed in a white brick wall by a mason. crystallized How came flint to be formed in layers? Is this layer of fint You may catechise yourself by applying similar series of If broken, what shattered it ? Below this layer of flint you found in every cliff of chalk? Is the flint whole, or broken? questions to any stone or pebble that you may meet, to granite, find two other lines of a dark brown colour. These lines run to slate, to coal, and to the different ores. parallel to each other, and keep about seven feet apart from This class of questions you may ask, any day, within twenty each other for many miles. They do not look like Aint: what yards of your own dwelling. You do not, however, always are they? They are seams of marl. What is marl? What is stay at home. You often take a walk or a ride. You some a seam. How came seams of marl into the chalk. As you whirl onward, you find that both the layer of flint and the times take a long journey; or you may make a short summer seams of marl break off suddenly, and as suddenly begin again excursion for business, recreation, or pleasure. On these oc- lower down in the cutting or section. How is this? Has casions you must, if you wish to become a geologist, .lways any portion of this rock ever sunk? or has some other portion take with you the same habit of personal observation, and the of it been thrown up, so as to disturb the continuance of these same system of asking questions. layers? If so, what force could have occasioned the dis turbance? On your journey from one part of the country to another you As soon as you find yourself through the MERSTHAM Tunnel find that the colour of the soil, as exhibited in the ploughed the daylight shows that you are in a completely new rock. fields, &c., differ very much from that of your own neighbour. What is that stone? Why is it called Firestone Has it any On hood. How and why is this? If you ride on horseback, or travel other name? Is it always found under the chalk ? by coach, you pass by gravel-pits, or rocks by the road-side, all leaving ReiGATE station you come to REDHILL. The hill on each hand consists of different coloured sands, layer upona of which are totally unlike those of the place whence you layer. What are these sands ? Why are they called Shanklin started. Try to account for this. What is gravel > were the sands? How is it that these sands are found here, at Shanklinn VOL. I. 10 in the Isle of Wight, Leighton Buzzard, and near Biggleswade, in Bedfordshire? Before you reach HORLEY you ride through flats abounding in clays and sands, which, as the cuttings show, furnish fuller's earth, and even iron ore. What is fuller's earth? What is an ore? How came iron to be formed in these sands? Is iron found in all sands? Why not? At HORLEY you come to a perfectly new series of rocks consisting of layers of clay and sand, and sandstones and shales. This group of rocks is called the Wealden. Why? What is shale? How came the sandstones at BALCOMBE to be, some in thin layers called laminæ, and others in thick masses called beds? What caused these beds to dip towards the north-east? But as soon as you pass the viaduct, you find that the very same beds dip towards the south-west. How is this? Did a force from below push up these beds till they snapped and then fell in different directions? By the Tunnel at HAYWARD HEATH, you see all these beds of clay, sand, shale, &c., exhibited in a deep cutting. In this neighbourhood you find Tilgate stone, called calciferous grit. What is grit? Why is this called calciferous? Is this like the rocks at TONBRIDGE WELLS and HASTINGS? How is it that the shales here look likely that coal would be found here: like those of coal? Is it Why not? At St. JOHN'S COMMON, you pass a rock that is crammed with shells. This rock is called Sussex marble. Did these shells ever live? What kind of shells are they? Are they like those in the sea, or like those found in rivers? If like those of rivers, is it likely that an immense river ever flowed in this district? While you are pondering these questions you come to STONEPOUND Gate, near HURSTPERPOINT, and you again enter the very sands which you left at REIGATE station. How is this? Did these sands ever form one continuous bed? If so, how came they to be separated? How came the rocks of TILGATE Forest to rise between them? And now ORSERVE that, as soon as you pass through these sands, you again enter the white chalk, as if you were going back from REDHILL to LONDON, instead of being on your way to BRIGHTON. You have now reached BRIGHTON. Look about you. Take a walk on the sea-shore near KEMP Town. Look towards ROTTING DEAN. Close to the water side you find a low cliff or bank of chalk. Resting on this wall of chalk you find a large and somewhat thick mass of loose soil, abounding with shingle, and large round stones, called boulders. What are these stones? Are they flints? No. Are thev granite, like the stones that pave the streets of London? If so, how came they to BRIGHTON? Where did they come from? The granite rocks nearest to Brighton are either in CORNWALL or at ABERDEEN in SCOTLAND. Did the sea bring these granite stones from CORNWALL? Will their colour help us to ascertain whether they come from CORNWALL, or from SCOTLAND? Did the present sea bring them? If so how is it that they are much higher than the high water-mark? Was there once a sea higher than the present? Was this the beach of that ancient sea? How did the sea change its level? Did the sea retire and sink? or did the land rise? How could this be?-But, look higher up. Resting upon this ancient sea-beach, you find a high cliff, consisting altogether of clay. What clay is this? Is it the same clay that you passed at New Cross? Why is it called the Elephant Bed? Did elephants ever live in this neighbourhood? If so, what was the climate of this region at that time? On your return to LONDON, put all these questions together, and try to obtain some intelligible truths out of them. Bethink yourself of the journey that you have made. You have begun and ended your excursion in a deep bed of clay, at NEW CROSS and at KEMP TOWN. You have passed through two rocks of chalk, one between CROYDON and MERSTHAM, and the other at CLAYTON-HILL, near BRIGHTON. You have crossed two beds of Shanklin sands, one at REDHILL, and the other near HURSTPERPOINT. You have travelled through two beds of what are called Wealden rocks, one near Hayward Heath, dipping to the north-east; and the other, near Balcombe, dipping to the south-west. Now, sum up all these hints of information, and compare the facts that you have marked. From New Cross to Hayward Heath, you have had, 1 Clay, 2 Chalk, 3 Shanklin sands, 4 Wealden rocks. From the viaduct near Balcombe, you have had again, 4 Wealden rocks, 3 Shanklin sands, 2 Chalk, 1 Clay. Here is perfect regularity: 1, 2, 3, 4 :: 4, 3, 2, 1. Hornblende, a dark crystalline substance, composed of alumina, fint, magnesia, and oxide of iron. Geognosy, though it does not occur in the lesson, is a name formery applied to the science of Geology; it comes from twe Greek words, ge and gnosis, which signify the earth and knowledge, and the changes which have taken place in its formation, untit it therefore means a knowledge of the earth as regards its structure, arrived at its present state. Of the two it is better to be a Geognostic than a Geologist, LESSONS IN FRENCH. some too. 4. The following idioms are followed by the preposition de SECTION XXI. when they come before a verb: (f 132) avoir besoin, to want ; 1. The expressions avoir besoin, to want ; avoir soin, to tako avoir coutume, to be accustomed ; avoir dessein, to intend, to care ; avoir honte, to be ashamed ; avoir peur, to be afraid, redesign; avoir envie, to have a wish, a desire; avoir honte, to be quire also the preposition de before a noun. Those idioms ashamed ; avoir intention, or, l'intention, to intend; avoir le mean literally, to have need, to have care, &c. temps, to have time or leisure ; avoir le courage, to have courage; Avez vous besoin de votre frère ? Do you want your brother avoir peur, to be afraid ; avoir raison, to be right; avoir regret, J'ai soin de mes effets. I take care of my things. to regret; avoir tort, to be wrong ; avoir sujet, to have reason ; Il a honte de sa conduite. He is ashamed of his conduct. avoir soin, to take care, Elle a peur du chien. She is afraid of the dog. Cet enfant a besoin de dormir. That child wants to sleep. 2. As these expressions require the preposition de before Vous avez honte de courir. You are ashamed of running. their object, they will, of course, require the same preposition before the pronoun representing that object. RESUME OF EXAMPLES. J'ai besoin de vous. I want you. Avez vous quelque chose à dire ? Have you anything to say? J'ai soin de lui. I take care of him. De qui avez vous besoin ? Whom do you want? Votre sour n'a-t-elle rien à écrire ? Has your sister nothing to write ? De quoi a-t-elle besoin ? What does she want? 3. When the object is not a person, and has been mentioned A-t-elle le temps de les écrire ? Has she time to write them 1 before, the prvnoun en takes the place of the preposition de, Elle n'a pas dessein de les écrire. She does not design to write them. Elle n'a pas l'intention de les écrire. She does not intend to write them. and that of the pronoun representing the object. Elle n'a pas envie de les écrire. She has no desire to write them. Avez vous besoin de votre cheval ? Do you want your horse ? I want it. J'en ai besoin. 4. The expressions être fâché, to be sorry; être étonné, to be Votre cousin a raison de sortir. Your cousin is right to go out. astonished; être content, to be satisfied, require the preposition N'avez vous pas soin d'écrire ? Do you not take care to write ? de before a noun or pronoun ($ 88). Avez vous le courage d'aller à la Have you the courage to go to the war? Je suis fâché de son malheur. I am sorry for his misfortune. guerre. Je suis étonné de sa conduite. I am astonished at his conduct. Je suis content de lui. I am pleased with him. 6. Etre fâché, in the sense of to be angry, requires the preAcheter, to buy. Faire, to make. Marcher, to walk. Champ, m. field. Fatigué, -e, tired, weary Mars, m. March. position contre. Danser, to dance, Gazette, f. newspaper. Ne-rien, nothing. Vous êtes fâché contre moi. You are angry with me. De bonne heure, early. Juillet, m. July. Page, f. page. 6. For rules on the government of adjectives, see $ 87, and Dormir, to sleep. Juin, m. June. Seize, sixteen. following Sections. Ecrire, to write. Lire, to read. Travaiiler, to work,labor. 1. Votre belle-mère a-t-elle quelque chose à faire ? 2. Elle Resume OF EXAMPLES. n'a rien à faire. 3. A-t-elle deux pages à écrire? 4. Non, Avez vous besoin d'argent ? Do you want money? Monsieur, elle n'en a qu'une. 5. Avez vous l'intention de lire J'ai besoin d'argent. I want money. I do not want any. cette gazette? 6. Oui, Madame, j'ai l'intention de la lire. Je n'en ai pas besoin (R. 3). 7. Avez vous raison d'acheter un habit de velours ? 8. J'ai En avez vous besoin ? Do you want any! raison d'en acheter un. 9. Votre petite fille a-t-elle besoin de J'en ai besoin, et mon frère en a I want some, and may brother wants dormir? 10. Oui, Monsieur, elle a besoin de dormir, elle est Avez vous besoin de votre frère ? fatiguée. 11. Avez vous peur de tomber ! 12. Je n'ai pas J'ai besoin de lui. * Do you want your brother I want him. peur de tomber. 13. Le jardinier a-t-il le temps de travailler De quoi avez vous besoin ? What do you want ! dans les champs ? 14. Il n'a pas envie de travailler dans les J'ai besoin d'un dictionnaire. I want a dictionary. champs. 15. Vos champs sont ils aussi grands que les miens? Avez vous soin de votre livre ? Do you take care of your book 1 16. Ils sont plus grands que les vôtres. 17. Avez vous honte J'en ai soin. I take care of it. de marcher 18. Je n'ai pas honte de marcher, mais j'ai | Avez vous soin de votre père ? Do you take care of your father ! honte de danser. 19. Quel âge a votre fils ? 20. Il a seize J'ai soin de lui.* I take care of him. ans. 21. Avons nous le deux mars ou le cinq juin ? 22. Nous Votre frère est il fâché contre moi ?) 18 your brother angry with me He is angry with your sister. avons le vingt-huit juillet. 23. Est il midi 24. Non, Mon. Il est fâché contre votre sæur. Are you afraid of this dog? Avez vous peur de ce chien ? sieur, il n'est pas encore midi, il n'est que onze heures et J'en ai peur. demie. 25. Il est encore de bonne heure. I am afraid of him. Of whom are you ashamed I Je n'ai honte de personne. I am ashamed of nobody. 1. What has your brother-in-law to do? 2. He has letters Je n'ai besoin de rien. Avez vous besoin de quelque chose ? Do you want anything! I want nothing. to write. 3. Does he want to work? 4. Yes, Sir, he wants to work. 5. Does he intend to read my book ? 6. He does EXERCISE 41. not intend to read your book, he has no time. 7. Is your Besoin, m. want, need. Fatigué, -e,weary,tired. Parler, to speak. sister ashamed to walk? 8. My sister is not ashamed to walk, Conduite, f. conduct. Gargon, m. boy. Reposer, to rest. but my brother is ashamed to dance. 9. Has your cousin Domestique, m. servant. Jeune homme, m. young Soin, m. care. Travailler, to work. anything to say? 10. My cousin has nothing to say, she is Effets, m. things, clothes. Vieux, old, afraid to speak (parler). 11. Is it late? 12. No, Madam, it Etonné, -e, astonished. Lire, to read. Fâché, -e, sorry, angry. is not sate, it is early, 13. Have you a wish to read my sister's letter (f.)? 14. Have you the courage to go to the war? 15. N'avez vous pas besoin de votre domestique ? 4. Oui, Mon 1. Qui a besoin de pain? 2. Personne n'en a besoin. 3. I have not the courage to go to the war. right to buy a silk dress (f.)? 17. Yes, Sir, she is right to sieur, j'ai besoin de lui., 5. Votre jardinier a-t-il soin do buy one, 18. Does that child want to sleep? 19. No, Sir, votre jardin ?. 6. Oui, Madame, il en a soin. 7. A-t-il bien that child does not want to sleep, he is not tired. 20. Has soin de son vieux père ? 8. Oui, Monsieur, il a bien soin de your brother's gardener a wish to work in my garden? 21. lui. 9. Votre gargon a-t-il honte de sa conduite ? 10. Qui, He has a wish to work in (dans) mine. 22. How old is that Monsieur, il en a honte. 11. Avez vous peur de ce cheval-ci child? 23. That child is ten years old. 24. What is the day id. 913. Notre domestique a-t-il soin de vos effets? 14. Il en of the month? 25. It is the ninth of March, 26. Are you a bien soin. 16. Avez vous peur de parler ou de lire! 16. Je afraid to walk? 27. I am not afraid to walk, but I am tired. 28. Have you time to read my brother's book? 29. I have n'ai' peur ni de parler ni de lire. 17. Etes vous étonné de time to read his book. 30. Has the joiner a wish to speak; cette affaire: 18. Je n'en suis pas étonné. 19. En êtes vous 31. He has a wish to work and to read. 32. Is your son afraid fåché? 20. Oui, Monsieur, j'en suis bien fâché. 21. Avez of falling? 33. He is not afraid of falling, but he is afraid of • The word en should be avoided as much as possible in relation to working? 34. What o'clock is it? 35. It is twelve. persons. man. vous besoin de ce garçon ? 22. Oui, Madame, j'ai besoin de lui. 23. N'avez vous pas besoin de son livre 24. Je n'en ai pas besoin. 25. Avez vous envie de travailler ou de lire? 26. Je n'ai envie ni de travailler ni de lire, j'ai envie de me reposer car je suis fatigué. remainder, and so on, whatever be the number of ciphers in the divisor. This principle is obviously the result of the decimal scale of notation; for if to place ciphers on the right of any given number is to increase its value in a tenfold proportion, then to cut them off will be to diminish its value in the same proportion. EXAMPLES: Divide 987654321 by 90; and, 142857100 by 7000. (1) Divisor 90) 98765432,1 dividend Quotient EXERCISE 42. 1. Do you want your servant? 2. Yes, Sir, I want him. 3. Does your brother-in-law want you? 4. He wants me and my brother. 5. Does he not want money? 6. He does not want money, he has enough. 7. Is your brother sorry for his conduct? 8. He is very sorry for his conduct and very angry against you. 9. Does he take good (bien) care of his books? 10. He takes good care of them. 11. How many volumes has he? 12. He has more than you, he has more than twenty. 13. What does the young man want? 14. He wants his clothes. 15. Do you want to rest (vous reposer)? 16. Is not your brother astonished at this? 17. He is astonished at it. 18. Have you a wish to read your brother's books? 19. I have a wish to read them, but I have no time. 20. Have you time to work? 21. I have time to work, but I have no time to read. 22. Does the young brother take care of his things? 23. He takes good care of them. 24. Is that little boy afraid of the dog? 25. He is not afraid of the dog, he is afraid of the horse. 26. Do you want bread? 27. I do not want any. 28. Are you pleased with your brother's conduct? 29. I am pleased with it. 30. Has your brother a wish to read my In the second example, you follow the same rule; that is you book? 31. He has no desire to read your book, he is weary cut off three ciphers in the divisor, and three figures in the dividend 32. Is that young man angry with you or with his frien is? and obtain the quotient as before, which is 20408, and remainder 33. He is neither angry with me nor with his friends? 3.1; to this 1, annex the three figures cut off from the dividend, and Do you want my dictionary? 35. I want your dictionary and you have the complete remainder 1100. The division may now be your brother's. properly represented thus: 90 LESSONS IN ARITHMETIC.-No. X. RULE OF SIMPLE DIVISION-(Continued). If the divisor to any proposed dividend happens to consist of any one of the nine digits, followed by one cipher, or any number of ciphers; then divide according to the following rule : Rule 2.-Place the divisor and dividend as directed in Rule 1, and mark off with a turned comma the cipher or ciphers contained in the divisor, and similarly an equal number of figures, to the right, from the figures contained in the dividend; then divide the figures of the dividend which remain to the left by the significant figure of the divisor, according to Rule 1. If there be any remainder when this division is performed, annex to it, on the right, the figures which were cut off from the dividend, and you will have the complete remainder which belongs to the complete divisor. If there be no remainder in dividing by the significant figure of the divisor, then the figures which were cut off from the dividend constitute the complete remainder belonging to the complete divisor. = The reason of cutting off the ciphers from the right of the divisor, and the same number of figures from the right of the dividend in this rule, is founded on the following principles:-1 If the divisor and dividend be both divided by the same number, their quotient will be unaltered; that is, it will be the same after this division as before. For example, if we divide 1728 by 8, we shall have the same quotient as if we divided each by 4, and then found the quotient of the results; because 1728-4-432, and 8-4=2, whence 432-2-216; but 17288-216 also, the same quotient as before. In like manner, 78,000 6,000 = 13; for 78 thousands divided by 6 thousands give 13 for the quotient. 2. If any number of ciphers or figures be cut off from a given number to the right, the figures which remain to the left form the quotient, which arises from dividing that given number, by a number composed of unity (or one) and as many ciphers are there in the number of ciphers or figures cut off. First, cutting one figure off a number to the right, is dividing it by 10; for example, to divide 3478 by 10 is to cut off one figure, thus: 3478, in which 347 is the quotient, and 8 is the remainder. Again, cutting two figures off a number to the right is dividing it by 100; for example, to divide 3478 by 100 is to cut off two figures, thus: 34,78, in which 34 is the quotient, and 78 is the remainder. Next, cutting three figures off a number to the right is dividing it by 1000; for example, to divide 3478 by 1000 is to cut off three figures, thus: 3478, in which 3 is the quotient, and 478 the • Repeat the preposition de. Quotient 20408-1100 remainder. In the first of these examples, you mark off with a turned comma, the cipher or 0 in the divisor, and the first figure 1, to the right, in the dividend; this is equivalent to dividing both divisor and dividend by 10. You now divide the remaining figures 98765432 to the left, in the dividend, by the divisor 9, according to Rule 1; thus, you obtain the quotient 10973936, and remainder 8; to this remainder, you annex the figure 1, which was cut off, and you have the complete remainder 81. The division may now be correctly represented, thus:987654321109739368 142857100 7000 In this second expression, the quotient of 1100 by 7000, which is the fractional expression of the remainder, is not altered in value by dividing both of its parts by 100, that is, by cutting off two ciphers from each. =20408 1100 70 EXERCISES. 1. Divide 123456 by 80; 876543 by 900; and 202037600 by 60000. 80000; and 428571428571-900000. nificant figures then proceed according to the following rule:When the divisor to any given dividend consists of several sig divisor by each of the nine digits, 1, 2, 3, 4, 5, 6, 7, 8, 9. This Rule 3.-Make a table of the multiples or products of the may be done either by multiplying the divisor by each of the digits separately, and tabulating the results; or it may be done by adding the divisor to itself, then to this sum, and again to this and each successive sum, till its product of nine times has been obtained by means is correct by the fact that when you add the divisor to the this addition. You will know whether the table obtained by this product by 9, the sum, if correct, will be the divisor itself, with a cipher annexed. You now mark the places for the divisor, dividead, and quotient, by placing on each side of the dividend a being generally placed on the left, and the quotient on the right of straight or curved bar, like an inverted parenthesis—the divisor the dividend; all this arrangement being merely matter of convenience and not essential to the operation of division. left of the dividend as are in the divisor; or, if the number comThe first step in this operation is to take as many figures to the parison with the table of its multiples which of them is nearest to posed of them be less than it, take one more, and ascertain by comthe number composed of these figures, which for the sake of brevity we shall call the dividuum; then, as this first dividuum contains the divisor as often as its multiple just found, put the digit which answers to the multiple in the quotient, and the multiple itself under the first dividuum; subtract the former from the latter, and to the remainder annex the next figure of the dividend, this will be the second dividuum; ascertain again by comparison with the table of multiples which of them is nearest to this dividuum: then, as this second dividuum contains the divisor as often as its multiple now found, put the digit which answers to this multiple in the quotient (that is, annex it to the former figure put in the quotient), and the multiple itself under the second dividuum subtract the former from the latter, and to the remainder ganes the next figure of the dividend; this will be the third dividuum; and with this proceed as before, continuing the process until the last figure in the dividend has been annexed; the remainder obtained at the last step will be the complete remainder, and the first figure in the quotient with those successively annexed will be the complete quotient. If, in the course of the operation, it be found, in any case, that when Divisor. Dividend. Quotient. Table of Multiples of the Divisor. 26 1 52 2 78 3 104 4 130 5 156 6 182 7 208 8 234 9 26 26 In the preceding operation it is useful to observe that as there are seven figures in the dividend after the first dividuum 34, which contains the divisor 1 time; therefore by annexing seven ciphers to this figure you have the number 10,000,000, which shows that the divisor 26 is contained in 340,000,000 ten million times. Again, as there are six figures in the dividend, after the next figure 7 in it has been annexed to the remainder 8, arising from the first dividuum, and as the second dividuum contains the divisor 3 times, therefore, by annexing six ciphers to this figure, you have the number 3,000,000, which shows that the divisor 26 is contained in 87,000,000 three million times. Consequently the divisor 26 is contained in 347,000,000, the first part of the dividend, 13 million times. In this way it is evident that we might proceed with the explanation, obtaining each figure step by step, and fixing its place in the quotient until the whole was explained in the same manner, and the proper quotient obtained, as shown in the preceding operation. MODE OF OPERATION. Here, as in the former example, you make a table of the mu tiples of the divisor, as exhibited on the margin. Having then marked out the places of the divisor, dividend, and quotient, you take the first five figures of the dividend, as the first dividuum ; comparing it with the table of the multiples of the divisor, you find the nearest multiple to be 60918; now, putting the corresponding digit 3 in the quotient, and this multiple under the first dividuum 62355, you subtract and find the remainder 1437; to this remainder, annexing 8, the next figure in the dividend, you find that the second dividuum 14378 is less than the divisor; that is, it does not contain the divisor even one time; you accordingly put 0 in the quotient; that is, annex 0 to the first figure 3 in the quotient, and proceed to find the third dividuum; by annexing the next figure 4 of the dividend, to the second dividuum 14378, which must now be treated as a remainder, you have 143784 as the third dividuum; comparing this with the table of the multiples of the divisor as before, you find the nearest multiple to be 142142; now, putting the corresponding digit 7 in the quotient, and this multiple mainder 1642; with this remainder, and the rest of the figures of under the third dividuum 143784, you subtract and find the rethe dividend, proceed as above, until you have exhausted all the figures of the dividend. You will thus obtain the quotient 3070809 and the remainder 430. Representing this division in the usual manner, we have MODE OF OPERATION. 62355847984 3070809 430 20306 Here you first make a table of the multiples of the divisor, by each of the nine digits, as exhibited on the margin. Having then marked out the places for divisor, dividend, and quotient, as directed, the first two figures 34 of the dividend on the left are found sufficient to contain the divisor, and are therefore called the first dividuum; on consulting the table of multiples, you find the nearest multiple to 34, to be 26; you therefore put the digit 1, which answers to this multiple in the quotient, and the multiple itself 26, under 34, the first dividuum; subtracting, you find the remainder 8; to this remainder, annex 7, the next figure of the The meaning of this expression is, therefore, that the number 20306 dividend, and you have 87, for the second dividuum; consulting is contained in the number 62355847984, so many as 3070809 the table of multiples again, you find the nearest multiple to 87, to be 78; you therefore put the digit 3, which answers to this multiple in the quotient; that is, annex it to the figure 1 already put in the quotient, and put the multiple itself 78, under 87, the second dividuum; subtracting, you find the remainder 9; to this remainder annex 6, the next figure of the dividend, and you have 96 for the third dividuum; with this proceed as before, until you have reached the last dividuum 163, which produces the last figure in the quotient 6, and gives the remainder 7. Hence, the complete quotient is 13371496 and the remainder 7; that is, the num-numbers from 1 to 49. ber 347658903 contains the number 26, so many as 13371496 times, with a remainder of 7. The division may now be correctly represented as follows : 430 times, and a fraction of a time denoted by 20306 EXERCISES. 1. Divide all the numbers contained in the first square on page 58, col. 1, No. 4, by 36; and the quotients will be all the numbers from 1 to 121. 2. Divide all the numbers contained in the second square on page 58, col. 1, No. 4, by 98998, and the quotients will be all the 3. Divide 10101010101 by 128; 437685 by 10, 100, and 1000; and 142857142857 by 999. : 3 4658903 =133714967 4. Divide the sum of all the numbers in the second square mentioned in Ex. 2, by the sum of all the numbers in the second square mentioned in Ex. 1. |