These visual rays must always be drawn from the extremities of lines, or any especial point which is to be represented in the picture, in the direction of the station point, or eye, but stopping at the picture plane (see Fig. 65); afterwards, from e, f, and g, they are drawn perpendicularly. For the reason why they are drawn perpendicularly, we refer the pupil to future lessons on geometrical perspective. Then produce or draw out one of the lines of the plan, say a c, to meet the picture plane. The point of meeting is called the point of contact, PC. Draw a perpendicular line from the PC to the base of the picture. We will call that PC 2, meaning the point of contact brought down. Join the PC2 to v P 2, and somewhere on this last line will be the picture of the object a c represented in the plan. This is determined by the visual rays being perpendicularly drawn to a2 and c2, therefore between a and c is the picture of the line a c; so, for the other line a b, draw a line from a2 to VP1, and the visual rays, as before, brought down, will determine the perspective length of a b-viz., a2 b2. Perhaps some add any more lines to that already given. We recommend the pupil to repeat the perspective view of the plan in Fig. 65, as given in Fig. 66. In this figure PC and P C 2 represent the points of contact of the line a c-that is, supposing the line were brought to the picture-in other words, to touch it. Then, in this case, it would be represented in the picture its natural size, therefore we call the perpendicular line drawn from PC to PC2 the line of contact, marked LC. Upon this line we always measure and set off heights of objects. Suppose, then, the height of the wall to be marked at r, draw a line from r to v P2; sto t will be the top of the wall ac; draw a line from s to v P 1; sm will be the top of the wall a b. Now if we wish to draw the courses of the bricks, we must set them off also upon the line of contact as we did to represent the top of the walls, and draw them to their respective vanishing points; also, the perpendicular joints of the bricks must be marked in the plan, and brought down by visual rays in the same way as the ends of the walls were found. We have represented a few of the bricks, leaving the reader may ask why we do not draw the line from PC 2 to v P 1, instead of v r 2. Our answer is, because PC is the point of contact for ac and not ab; if ab had been produced to the PP for a point of contact, then it would have been right to draw a line from PC 2 in the direction of v P 1. All that we have now done in this perspective diagram is, that we have shown the horizontal retiring length of the base of the wall each way-viz., a2 c2 on one side, and a2 b2 on the other. To have drawn these lines equal to the length of the walls themselves that is, those of the plan-would have been a very great mistake, because as they retire the further pupil to complete the drawing; the plan of the door is shown at no, its height at p. (We will observe, by way of parenthesis, that all heights of objects are marked or set off on the line of contact; all horizontal lengths and breadths are shown in the ground-plan, and brought down by visual rays.) We will give one other method of showing the horizontal perspective length of a line or plane, and then leave the pupil to think over and prae. tise all that we have been trying to teach him. Let a b (Fig. 67) represent the length of a line to be shown in perspective at a given angle with our position or with the picture plane. Let PS represent the point of sight, sp the station point, H L the Let horizontal line or height of the eye, BP base of picture. the period, and as many ciphers as there are figures in the nonrecurring part. 25. It will be seen from the above detailed explanation of the method by which the equivalent vulgar fraction may be deter mined, that an analogous method would apply to any circulating decimal whatsoever. Hence we get the following Rule for reducing a Circulating Decimal to a Vulgar Fraction. Subtract the number formed by the figures of the non-recurring part from the number formed by the figures taken to the end of the first period, and set down this difference as a numerator. Take as many nines as there are figures in the period, and, annexing to them as many ciphers as there are figures in the non-recurring part, set down the number so formed as a denominator. 26. We have proved the rule in the case of a mixed circulating decimal. The case of a pure circulating decimal is included in it; for in a pure circulating decimal there is no non-recurring part, and therefore nothing to be subtracted, and the denominator will consist wholly of nines, their number being equal to the number of figures in the period. Thus 67 = 83, 053 = 4. 27. For the sake of clearness, however, we will perform the process for a pure circulating decimal. Take 67, for instance. Let, as before, ƒ 676767. . . . ; Then, 100 f = 67 676767 ....., 99 ƒ = 67, Or, f = 1; Having now shown, as we promised, how the retiring horizontal distances of objects may be faithfully represented on paper, we will give some examples as subjects for exercises. Fig. 69 is an example of a retiring row of posts, their distances being purposely shown by the geometric method of the last two and therefore subtracting, as in the previous case, problems. It is almost needless to direct the attention of the pupil to the diminishing retiring spaces between the posts; however, he will see, as we have previously endeavoured to make clear to him, that those retiring distances can be satisfactorily proved. Fig. 70 is given as an exercise, including many of the principles we have before explained-viz., angular perspective, horizontal retiring lines, inclined lines of the roofs, and horizontal retiring distances, all of which the pupil, we trust, will now be able to arrange for himself, and to find his vanishing points. LESSONS IN ARITHMETIC.—XVII. 24. To reduce a given Circulating Decimal to a Vulgar Fraction. Denote the true value of the equivalent fraction by f. Then f = 34567567567 ....., the period 567 being supposed continued ad infinitum. If we multiply f by 100000, and also the decimal by 100000, the results will still be equal. Hence 100000 ƒ = 31567-567567567 .... The decimal place being moved five places to the right, and the period 567 being still continued ad infinitum on the right of the decimal point as before. Similarly, 100 f = 34.567567567 ..... Now observe carefully how each part of this fraction has arisen. The numerator is obtained by writing down the figures of the decimal as far as the end of the first period without the decimal point, and then subtracting from the number so obtained the figures which occar before the period, or, as we may call it, the non-recurring part. The denominator 99900 arises from subtracting 100 (i.e., 10 raised to the same power as the number of figures in the non-recurring part) from 100000 (i.e., 10 raised to the same power as there are figures in the non-recurring part and period together). This subtraction will necessarily produce a number 99900, containing, that is to say, as many nines as there are figures in and it is evident, from the way in which they arise, that the number of nines in the denominator is equal to the number of figures in the period. 28. Of course, if there is an integral part in the original decimal, that will remain unaltered, and the required answer will be a mixed number, which may be reduced to an improper fraction if necessary. EXAMPLE.-3.1415. = 1415141401 9900 0900 Taking the decimal part separately, 1415 Hence 3.1415 31101 = 11101 3900 expressed as an improper fraction. Or it may be expressed as an improper fraction at once:3.1415 = 21415314 31101 9900 The truth of this latter method may be established exactly in the same way as the two cases we have already explained. 29. The learner is recommended at first, in reducing circulating decimals to vulgar fractions, to perform the operation in the way we have indicated in the examples already given-i.e., by multiplying by the requisite powers of 10, subtracting, etc. He will thus better appreciate the truth of the rule which he will afterwards employ. It is evident that the equivalent fractions found by the rule will often not be in their lowest terms. EXERCISE 35. Reduce to their equivalent vulgar fractions the following decimals 1. .3. 2. 03. 5. 2349. 9. 27.5238. 6. 42623. 10. 21-000008. 13. 052100. We thus see that the difference between the decimal and the true value of the fraction continually diminishes. In the case of a terminating decimal this difference becomes zero when we have taken all the figures in. In the case of a circulating decimal, it never actually becomes zero, but we can make it as small as we please by taking a sufficient number of decimal places. 31. When a result is required correct only to a certain number of decimal places, it is better, as we have already explained (Art. 14), to find one figure more of the result than is actually required, so as to ascertain whether this figure is greater or less than 5. If it is greater, we increase the figure in the last place which is required in the result by 1. The following is an example of a decimal continually approximated to in this way, by taking successive figures, and increasing, where necessary, the last figure by unity Let 4:89169 be the decimal. The successive approximations would be5, 49, 4'89, 4-892, 4 8917, 4'89169. Here 5 is nearer to the true value than 4 would be. 32. Operations in which circulating decimals occur are better conducted by reducing the circulating decimals to their equivalent vulgar fractions, if absolute accuracy is required. approximate result is desired true to a certain number of decimal places, then, in additions and subtractions, it will be sufficient to take in two or three figures of the period beyond the number of places required, and then add or subtract. For instance, in adding 4567 to 312468) correctly to 9 decimal places, we should write the decimals as follows: 1. 3. 6. 72. 11. 16. 16. .583. 2. 6. 3. 18. 7. 09. 12. -8567923. 17. 0227. 13. 138. 18. 4745. 14. 53. 19. 5925. 4. 123. 9. 142857. 5. 297. 10. 076923. 15. 5925. 20. 008497133. LESSONS IN ENGLISH.-IX. DERIVATION: PREFIXES (continued). BEFORE proceeding further with these prefixes, we may now expose a common error. It is generally thought that words have several disconnected significations. Several significations many words have, but these significations are all allied one with another, and they are allied one with another in such a way that a genealogical connection runs through them all. I mean that the second ensues from the first, and conducts to the third. The meanings of words flow from a common source, like the waters of a brook. That common source, or parent-signification, is, in all cases, one that denotes some object of sense, for objects of sense were named before other objects. Our first duty, then, is to ascertain the primary meaning of a word. From that meaning the other meanings flow, as by natural derivation. Those secondary or derivative significations, then, can scarcely be termed meanings; they are not so much meanings as modifications of the primary import of the root. Certainly they are not independent significations. Thus viewed, words have not two or more senses, but in the several cases the one sense is varied and modified. Even in instances in which opposite meanings are connected with the same word, the filiation may be traced, as both Jacob and Esau sprang from the same stock. I will take an example in the word prevent. Prevent means both to guide and to hinder, to lead to, and to debar from. The opposition is sufficiently decided. Yet these two opposed meanings are only modifications of the root-sense of the word. First I will exhibit the diversity, and then explain it. Prevent, signifying to guide, aid forward :— "Prevent us, O Lord, by thy grace."-" Book of Common Prayer." "Love celestial, whose prevenient aid Forbids approaching ill."-Mallet. Prevent, signifying to hinder, obstruct: "Where our prevention ends, danger begins."-Carew. "Which, though it be a natural preventive to some evils, yet without either stop or moderation, must needs exhaust his spirits."-Reliq. Wottoniana. "Physick is either curative or preventive; preventive we call that which preventeth sickness in the healthy."-Brown, "Vulgar Errors." "Prevent us, O Lord, by thy grace," means "aid us forward." "Preventive of sickness," signifies that which causes sickness not to come. There is the contrariety. Now for the explanation. Prevent is made up of two Latin words, namely-præ, before, and venio, I come or go. Now, you may go before a person for two opposite purposes. You may go before him in order to guide, aid, and conduct him onward; or you may go before him to bar up his way, to hold him back, to prevent his advance. And as either of these two purposes is prominent in the mind of the speaker, so the word is used by him to signify 8. 5301-357 + 72-39 + 187.21 + 4·2965 + 217-8196 + 42-176 + 523 to guide or to hinder. The proper meaning, then, of prevent is, 58:30048. to come before: hence, 1, to guide, or, as a natural consequence, 2, to aid; or again, 1, to obstruct, and, as a natural consequence, 9. 182 + 134-09 + 2·98 + 97·26 + 3·769230 + 99.083 + 15 +814. 2, to stop, etc. And how the moral and spiritual imports come out of the physical, is also seen in the diverse application of the word; for, as we have just read of preventive medicine, so in divinity you may read of "prevenient grace.” These remarks, illustrations of which occur in what has just preceded, and will occur in what is about to follow, may serve to show you that language must be studied genealogically. Indeed, every word has a history; and in the dictionaries, every account given of a word ought to be a complete history of the word; a history of its origin, uses, and application, the one traced from the other logically, or according to the laws of thought, and philologically, or agreeably to the laws of language. Very different, and very inferior, is the character of most dictionaries. But to return to the subject of English prefixes. E, of Latin, or rather Greek origin, in the forms e, ef, ex, denotes out of, as in egress (e and gradior, Lat. I walk), a walking out; excess (ex and cedo, Lat. I go), a going beyond—that is, too far; effect (ef and facio, Lat. I do), a thing made out, produced; a result. E. "All occasions must be taken of sending forth pious heavenly ejaculations to God."-Bishop Hall. Ex. "The ecclesiastical courts possessed the power of pronouncing excommunication; and that sentence, besides the spiritual consequences supposed to follow from it, was attended with immediate effects of the most important nature. The person excommunicated was shunned by every one as profane and impious; and his whole estate, during his lifetime, and all his movables, for ever were forfeited to the crown."-Hume, "History of England.” Ef. "Two white sparry incrustations, with efflorescences in form of shrubs, formed by the trickling of water."-Woodward, "On Fossils." En is a prefix found in the English, the French, and the Greek languages. Into the English it appears to have come from the Latin, through the French. Many words of Latin origin have passed through the French into the English. En is the form in Greek. In Latin, en becomes in. In French, both en and in are used. The same is the case with the English. Though en and in are the same particle, it may be advisable to handle them separately, in order that their respective usages may become apparent. En is found in the forms en, em. The prefix signifies in or into, e.g.: Not unto such as could him feast againe, And double quite for that he on them spent; But such as want of harbour did constraine, Those, for God's sake, his dewty was to entertaine." Spenser, "Faerie Queene." Epi, a prefix of Greek origin, from eri (ep'-i), signifying upon, as epidemic, upon or over (widely spread over) a people. En demic declares that a disease is in-born, native to the soil; epidemic that it is very prevalent. Epi is found in epigram (epi and the Greek ypauua, pronounced gram'-ma, a writing, from the verb ypapw [graph'-o], I write), epilepsy (epi and Ania, pronounced leap'-si-a, a taking), epiphany (epi and Greek pairo, pronounced phai'-no or fi'-no, I appear), epistle (epi and σreXXW, pronounced stel'-lo, to send), etc. etc. "He that would write an epitaph for thee, Thy worth, thy life, but he that hath lived so."-Donne. Equi, of Latin origin (æquus, equal), denoting equality, forms part of several words, as equipoise (equi and peser, Fr. to weigh; pendere, Lat. to hang), equity; equivocal (equi and vox, Lat. a voice). "Faith! here's an equivocator that could swear in both the scales against either scale; who committed treason enough in God's sake, yet could not equivocate to heaven; oh, come in, equivocator."--Shakespeare, Macbeth." Es, of French origin (Lat. e, ex), is in English found in words borrowed from the French, as in escalade (es and scala, Lat. a ladder), a scaling (of a city), escape (Fr. échapper, to get away), escheat (old Fr. escheoir, to fall due), a forfeit, eschew (old Fr. eschever, to shun), escutcheon (es and scutum, Lat. a shield). "Hence without blushing (say whate'er we can) The hound or spaniel as his talent lies."-Cawthorn. Eu, of Greek origin (ev, pronounced you), signifying well, occurs in euphony (eu and the Greek pwvn, pronounced pho'-ne, a sound), euthanasia (eu and the Greek @avatos, pronounced "He (Samson) rises and carries away the gates wherein they thought the word; eunuch being from the Greek evvn, pronounced n'-ne, than'-a-tos, death), a happy death; the eu in eunuch is a part of to have encaged him."-Bishop Hall. So in encamp, encase, enchain, enchant, enclose (or inclose), endemic (en and demos, Gr. a people), peculiar to a district. En sometimes has an intensive or augmentive effect on the verb of which it forms a part; as in encourage, enfeeble, enkindle (candle), encrease (increase), encumber (incumber, from the French encombre, Lat. cumulus, a heap). "Encumbered soon with many a painful wound, Rowe, "Pharsalia," En has also, though seldom, the force of a negative; as in enemy. Enemy is from the Latin inimicus, where the English en represents the Latin in. Inimicus is made up of in, not; and amicus, a friend. En, for the sake of euphony, becomes em before b and p; embitter, emblem, embosom, embroil, emprison (imprison), employ, empoverish (impoverish). "At eve within yon studious nook, I ope my brass-embosséd book, Of martyrs crowned with heavenly meed."-Warton. There is a tendency to substitute i for e in many words. This tendency deserves encouragement, if only for the sake of uniformity. Enter, coming from the Latin (intra, within) through the French (entre, between, among), is found in enterprise (enter and Fr. prendre, Lat. prehendere, to take, take hold of), an undertaking; also in enterment (in and terra, Lat. the earth), now more common as interment. It is found also in entertain (Fr. entretenir, Lat. inter and tenere, to hold). "His office was to give entertainment And lodging unto all that came and went, a bed, and exw, ek'-o, to have, or have charge of; eunuchs were chamberlains. Men were made eunuchs by the jealousy of Eastern despots. They were also made so in order to give them a contralto voice. The latter fact is well alluded to in this quotation: "Our present writers, for the most part, seem to lay the whole stress in their endeavours upon the harmony of words; but then, like eunuchs, they sacrifice their manhood for a voice, and reduce our poetry to be like echo, nothing but a sound."-Lansdown, "Peleus and Thetis." Ever, of Saxon origin, signifying always, is seen in everlasting, evermore; evermore appears in the older writers as evermo. "I shall readily grant that the words for ever and ever-lasting do not always, in Scripture, signify an endless duration." — Barrow, "Sermons." Extra, of Latin origin, with the meaning out of, appears in extraneous, out of (not belonging to) the subject; extraordinary (extra and ordo, Lat. order), out of the usual order. "Some lands, either because they were in the hands of irreligious and careless owners, or were situate in forests and desert places, or for other now unsearchable reasons, were never united to any parish, and therefore continue to this day extra-parochial."-Blackstone, “Commentaries." For, of Saxon origin, whose original is probably found in the German ver, which denies and reverses the action expressed in the verb, occurs in forbid (not to bid; that is, to bid not). "Rather how hast thou yielded to transgress, The strict forbiddance, how to violate The sacred fruit forbidd'n."-Milton, "Paradise Lost." For is found also in forbear, not to bear or take; to abstain. "Phidias, when he had made the statue of Minerva, could not for bear to engrave his own name, as author of the piece."-Dryden. Fore, a different word from the preceding, of Saxon origin (vor, RECREATIVE NATURAL HISTORY. Germ., in advance; vorwarts, Germ., forwards), appears in foretell, forecast, forefathers, forehead. "The foreknower is not the cause of all that are foreknown." Hammond. In forgive (Germ. vergeben), the idea seems to be that of giving away, giving without a return, giving freely, and hence to pardon (Fr. pardonner, in low Lat. perdonare). "Not soon provoked, however stung and teased, And if perhaps made angry, soon appeased; She rather waves, than will dispute her right, And injured makes forgiveness her delight."-Cowper. Hept, of Greek origin (errα, pronounced hep'-ta, seven), forms the first syllable of heptagon (Greek yovia, pronounced gon'-i-a, an angle), that which has seven angles, and consequently seven sides; and heptarchy (Greek apxn, pronounced ar'ke, government), a sevenfold government. "Seven independent thrones, the Saxon heptarchy, were founded by the conquerors.”—Gibbon. RECREATIVE NATURAL HISTORY.-I. THE SNAIL. It is to be feared that there are not many among us who are disposed to regard the little animals that may be classed among the "common objects" of our fields, gardens, and even houses, with the same attention and curiosity as we examine the form and inquire into the habits of a lion, elephant, or gorilla, fresh from the deserts of Africa or the jungles of Asia, or a walrus lately brought from northern climes. And yet the beasts that find a hiding-place in our woods and thickets, the birds that fill the air with melody at the approach of spring, and the insects that often destroy our best and choicest fruits and blossoms, are as "fearfully and wonderfully made' as the larger animals of foreign lands-ay, even as ourselves, for whose use, or pleasure, or perchance correction, they were created. Each has been called into being for some wise end by the Maker of us all, even though our limited knowledge may fail to discover its utility, and the purpose which it serves in the economy of Nature. The structure and habits of each beast or bird or insect, however small, however unattractive in appearance, claim our consideration as much as the graceful figure of the antelope or giraffe, or the instinct and docility of the horse or dog; and as a lesson may be learnt from each and all, more potent in its teaching than the precepts of the best of all books save one, we invite the attention of our readers to our studies in Natural History, which may be termed recreative in two senses-first, as they will do much to relieve the strain that our lessons in languages, mathematics, and science may exert on the mind of the student; and secondly, in the first and truest meaning of the word, as by a thoughtful inspection of some of God's lesser works, we may renew from time to time and build up again what we may have lost of our reverential love of Him without whom not even a sparrow falls to the ground unnoticed or uncared for. In such a spirit, then, we introduce to the notice of our readers the snail, an animal that finds small favour, generally speaking, with those who love their gardens. THE SNAIL. We will imagine that while strolling round your garden or in the fields you have just picked up a snail. Hold him tenderly, and not long in your hand, or you may make him very wretched. How so? Remember his body is cold, your hand is hot, almost like a furnace to him, and the temperature must be enough to make him faint. In truth, while on a human hand the snail must feel about as comfortable as St. Lawrence on his gridiron. Besides, St. Lawrence gained honour and applause for his suffering, but no such reward awaits the snail; so, out of a kindly feeling, do not keep him long in the hot hand. 269 Then how shall we observe our friend and study his comfort also? Get a piece of clean window-glass, and place the snail upon it. He will hold firmly to the glass with his broad, expanded, sucker-like foot. Then, by looking at the gentleman through the glass, as he moves along, the reader will be able to note the mode in which such animals walk, mark the wave-like motions of the foot on the glass, and remember that all softbodied animals with a foot like the snail's, are named Gasteropods, a word which means "having the feet and belly joined," and which is derived from the Greek yaoтnp (gas-teer'), the belly, and wous (pous), a foot. Having noticed the sucker-like foot, and tested the force with which it clings to the glass, let us look at the head of our snail. The first noticeable objects are what children call the horns or feelers. Look closely at them. What is that black shining speck on the top of each feeler? The eye of the snail, according to the judgment of most naturalists. Strange sort of eye, which can thus be lifted up above the body, when its owner wants to take a survey of the world. If we want to obtain a wider view, we get on an elevation; the snail manages matters in another manner, he lifts up the eye itself. As the snail contemplates one of us through those black specks, the question rises, is he not terribly frightened at a being having an eye as large as his whole body? However, unfortunately, in the present state of snail education, it is impossible to impart his views to us, so we will let that topic pass. Touch the tip of his feeler; see how ingeniously he tucks the whole machine into its case, just as the top of the finger of a glove is turned in sometimes, when the glove is drawn off. Now wait awhile; see, the tube is pushed out again, and the eye is slowly rolled out from its remarkable hiding-place. Have you a pair of scissors in your hand? Would you like to cut off those feelers, eyes and all? No, some will say, respect even a snail's feelings. Others may answer yes, cut them off, if we shall get any knowledge by so doing; we do not believe such creatures feel pain. Well, you cannot prove they do not feel when thus treated, that's certain; and it shows a better heart to believe they suffer when injured. Those who believe in Shakespeare will probably take this view. They will remember his remark that a worm when crushed feels as much pain as when a giant dies. However, let us suppose one of us determines to be rather cruel for once only; that we do violence to our tender feelings, and, disregarding the snail, cut off both of his feelers. Now we have killed him, have we not ? At least we have blinded him for life? Indignant the snail is certainly; see how he goes back into the innermost part of his house. He may well retire from a world which treats him thus. Now what will be the result? If the snail be in good health, and the operation be not performed too late in the year, that poor despicable-looking creature will begin to form a new pair of eyes and feelers in about twenty-five days. This operation was often performed on a great number of snails by Spallanzani, a celebrated Italian naturalist of the last century. Such a reproduction of organs proves the possession of singular vital powers in so lowly a creature. But Spallanzani and others have gone beyond this. They repeatedly cut off the heads of snails, and those heads, with all their organs, have been in a few months reproduced! That is a power which some men might have envied. Even the little finger of a human being when cut off is gone for life; no power of making a fresh one grow on the old place belongs to the greatest philosopher on the earth. Yet here we have a poor despised creature often able to recover its lost head, eyes, feelers, and mouth. The snail beats us all on such a work, beyond doubt. Let us not forget the mouth of the snail. It is an instrument capable of doing no light work, as those know to whose gardens the animal pays its unwelcome visits. The two lips are formed of a horny substance, which acts in the manner of a file on vegetables. The tough leaves of the white lily are often rasped off in a few nights by this cutting machine. If any one should be desirous of examining minutely the structure of the snail's mouth, he will find some fine specimens in the Physiological Gallery of the Hunterian Museum in the College of Surgeons, Lincoln's Inn Fields. Of the snail's brain we may just make this remark, that the complete nervous system of the creature's first cousin, the slug, is to be seen in the same museum, and Professor Owen has given a learned description of the whole. Both snails and |
