a fleet; messis, a crop of corn; clavis, a key; navis, a ship. The ablative singular has for the most part i (perhaps from ie) instead of e in parisyllabics with the vowel-stem in i. In imparisyllabies with consonantal stems, e is the usual ablative termination, but is sometimes found, derived from the usage in the vowel-stems. Nouns which make the ablative singular in i, make the genitive plural in ium instead of um; and nouns neuter, which in the ablative singular end in i, in the nominative, accusative, and vocative plural end in ia. Adjectives of the third declension, in general, follow the declension laws of the nouns, only that in the ablative singular they prefer i. Adjectives of the third declension are of two sorts: first, those that have three terminations, as, alucer, m., alăcris, f., aluere, n., lively, active; second, those that have two terminations, as the comparative, vilior, m. and f., vilius, n. meaner; under this second class may stand such as ferox, fierce, which in the nominative singular is r., f., and n. (accusative, ferocem), but in the plural has for the neuter a separate form in ie, as ferocia. DECLENSION OF AN ADJECTIVE OF THREE TERMINATIONS. EXAMPLE.-Acer, acris, acre, sharp, acute, pungent, energetic. Singular. Cases. M. F. N. acris. acre. G. acris. acris. acris. D. acri. acri. Ac. acrem. acrem. acri. V. acer. Ab. acri. acris. acre. LESSONS IN DRAWING.-IX. THE aim of all instruction in drawing ought to be, first, to convey in as clear and simple a manner as possible the best means of judging of the relative proportions of objects, not only with regard to their individual component parts, but also with reference to the proportions these objects bear to one another; and, secondly, to place before the pupil the most ready methods of representing these objects, subject as they are to an enlless variety both of form and position." How is it that when standing upon the side of a hill, and looking over a large extent of country, if we raise the hand and hold it parallel to our eyes at arm's length, it will cover or prevent our seeing probably many miles of landscape, including houses and villages? Or, if we select a closer object-for instance, the house on the opposite side of the street-and place the hand as before, we find the result to be the same? Simply because as objects retire, or are further from the eye, they occupy less space upon the vision than when nearer. Here then, we have practical evidence that to represent these objects correctly we must inquire for some means which will enable us to accomplish our task, and satisfy our minds that we have given these objects their right proportions as they retire, and that each object, and cach part of an object, occupies its proper space upon the paper as it does in the eye; in short, giving them their true scale of representation according to their distances from ourselves and from one another. The science of perspective enables us to accomplish this end, and although we do not attempt, in these lessons upon free-hand drawing, to go very deeply into geome trical perspective, yet we find it absolutely necessary to make some use of it in order to render our explanations clearer; for by the assistance of rules, difficulties are lessened, especially when we can classify many objects and the circumstances in which they are placed under the same principles. We said in a previous lesson that there were rules in perspec. tive for regulating the retiring horizontal distances of objects, as well as their heights; and we now propose to give such of these rules as are absolutely necessary for the pupil's guidanco in free-hand drawing. We must first remind the pupil of what majoribus, majoribus, majoribus. has been already said respecting the theory of planes or surfaces. Audax, m. and f.; audacia, n., bold. V. major. major.. majus. Ab. majore. majore. majore. Singular. Cases. M. and F. audacia. audacium. audacibus. audacia. audaces. audacia. audacibus. audacibus. KEY TO EXERCISES IN LESSONS IN LATIN.-VIII. EXERCISE 25.-LATIN-ENGLISH. 1. I have great grief. 2. Hast thou not great grief? 3. Mothers have great griefs. 4. The colour of the cushion is beautiful. 5. Is the colour of the cushion beautiful? 6. He has (is under) a deadly error. 7. Why has father (is under) deadly errors? 8. I have a brother. 9. Brothers have great griefs. 10. Lightning frightens animals. 11. Does not lightning frighten mothers? 12. Lightning frightens sparrows. EXERCISE 26.-ENGLISH-LATIN. 1. Est mihi calcar. 2. Estne tibi auser? 3. Illis sunt anseres. 4. Estne tibi agger? 5. Fulguris odor in pulvinari est. 6. Vectigalia 1. I fear charcoal. 2. The boy strikes the peacocks. 3. The regions are beautiful. 4. Thou hast an opportunity. 5. We move the ashes. A horizontal plane is a plane parallel with the earth; a perpendicular plane is one perpendicular to the earth. The top of a table and the ceiling of a room are horizontal planes; the walls of the room are perpendicular planes. These are visible planes. We are sometimes, in practical perspective, compelled to uso imaginary planes. These more properly belong to the practice of geometrical perspective. It will be very necessary for the pupil, if he wishes thoroughly to understand the principles of drawing objects at a given distance from him, especially buildings, to go very attentively through future lessons on geometrical perspective, given in the pages of the POPULAR EDUCATOR, for this reason: no one ought to be satisfied with the result of his work, even if it be correct, unless he knows the whole of the why and the wherefore which have brought out the result. It is, unfortunately, a very common practice in some books of instruction upon drawing, when the subject is a building, to mark a copy with letters-a, b, c, d, etc.—and carry the instructions no further, but merely tell the pupil to draw from a to b, and from c to d, and to observe that d is a little higher or a little lower than c, as the case may be, without any mention whatever as to why d should be higher or lower. Now in this, and all similar cases, a little knowledge of perspective would make the practice simpler and the result certain. Tho pupil may make an exact imitation of his drawn copy, but that is not enough; he must be able to do the same from the object; and how is this to be done correctly by such a system as that which only enables a pupil, parrot-like, to reproduce a copy and nothing more? But we hope that very few of our readers will like to stop there. To draw from nature and the real thing, we trust, is the ambition of every one who makes up his mind to go through these lessons, that he may make the art of drawing a useful and valuable auxiliary to his occupation as a means of expressing himself, as well as a pleasing recreation for leisure hours. Another reason why we recommend the pupil to study our lessons in geometrical perspective is, as we have said before, when treating upon drawing a simple outline from the flat (a term used by draughtsmen when copying from a drawing), that the practice of geometrical perspective assists the eye to under panying barns, stables, strawyards, etc. etc.-that we must first make a measured plan of the whole, and go through the drawing geometrically, before we can hope to make a truthful picture. It would be as ridiculous to suppose that when we write a letter or an essay, we ought to repeat all the rules of syntax, so that the grammatical construction of the sentences may be correct. Every educated man knows that the right words flow naturally into their places in proper agreement and sequence. The phrases harmonise without any effort on his part, simply because he knows the rules, and experience makes them easy to apply. stand and calculate more readily the proportions of retiring lines and planes. As a practical illustration of this principle, we meet with it repeatedly in the readiness with which an experienced carpenter will tell you the length of a board without taking the trouble to measure it. His eye is so accustomed to the foot-rule, and the space a repeated number of measurements will cover, that to him it is no difficulty to say within a very close approximation how long the board is. It is the repeated practice of geometrical perspective that enables a draughtsman to decide upon the proportional length of a line or plane as it retires, and to draw either correctly on his paper. If we did not consider it in this way with regard to free-hand drawing, it would be of very little use in the practice of drawing from nature. It would be absurd to expect, when we are seated before a subject-say a picturesque farmhouse, with the accom We will now give a geometric method of representing two walls meeting at an angle, as an illustration of what we have stated. Let two lines, ab, a c (Fig. 65), forming an angle of 90 degrees, represent the plan of two walls meeting at the point a, of which ba forms an angle of 40 degrees with the picture plane. PP is the picture plane, H L the line of sight, BP base of the picture, s P the station point, and VP1 and VP2 are the vanishing points for the corresponding numbered lines of the plan. First draw the picture plane, and then the line ba, placing it at an angle of 40 degrees with the PP; then from a draw a c at an angle of 90 degrees-that is, a right angle-with ab; this will be the plan of the walls as they are placed before our vision. Then mark SP to represent the supposed distance we are from the angle of the walls. Find the vanishing points for the two lines of the plane. We have already given the rule 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 VP 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; o, for the other line a b, draw a line from a to VP1, and the visual rays, as before, brought down, will determine the perspective length of a b-viz., a 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, aя 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 VP1; 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 P 2. Our answer is, because PC is the point of contact for ac and not ab; if a b 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 prac 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, s P 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-recur. ring 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 circulat ing 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 = $1,053 83 27. For the sake of clearness, however, we will perform the process for a pure circulating decimal. Take '67, for instance. Let, as before, f =676767. . . .; Then, 100 f 67*676767 .... 9 99 f = 67, Or, f 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. -34; 99900 34532 Hence 99900 f = 34567 and f, the fraction required, =34%07 —31 = 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 occur before the period, or, as we may call it, tho 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. = 9900 9900* Taking the decimal part separately, 1415 = 1415-14 = 1403 Hence 3.1415 31401 31101 300 expressed as an improper fraction. Or it may be expressed as an improper fraction at once:3.1415141514 = $120). 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. 3. 032. 5. 2349. 9. 27.5238. 10. 21.000008. 13. 052100. 14. 181-032416. 11. 52-314159. 4. ·523. 8. 357-003129. 30. Approximation. places, etc. 15. 0000549. 16. 612512527. Decimals correct to a given number of We have already remarked, that if we take only a limited number of the figures of a decimal, we approach nearer and nearcz to the true result as we continue to take in more figures. We give an example, taken from De Morgan's "Arithmetic," which shows this clearly. = 142857 a circulating decimal. Now taking successively one, two, three, etc., figures of the decimal, we haveis less than by دو which is less than 1498 10000 34285 100000 14བས་༡ 1000000 دو We thus see that the difference between the decimal and the In the case of true value of the fraction continually diminishes. 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 4-99169 be the decimal. The successive approximations would be5, 49, 4'89, 4892, 48917, 4.89169. is nearer to the true value than 4 Here 5 4.9 would be. 4.8 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. If an 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 3124689 correctly to 9 decimal places, we should write the decimals as follows: 1. Write down the decimals containing respectively one, two, three, four, five, and six decimal places which are the nearest approximation to the decimals 67819473, 203781947. DERIVATION: PREFIXES (continued). BEFORE proceeding further with these prefixes, we may now 2. Find the value correctly to seven decimal places of the more senses, but in the several cases the one sense is varied and following expressions : 1. 20127 + 89-3897 + ·003701. 2. 15-379 + 2·13459 +18 + 70-2178 + 5.34567. 8, 27-459 - 3·876139. 4. 7-28705 - 378 + 10-31567. 21 + 5·123 – 2·315. 6. 31 2:39 + 3.28. 7. 3-6 + 78-9176 + 735·3 + 375 + ·27 +187·4. modified. Even in instances in which opposite meanings are 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 explanetion. Prevent is made up of two Latin words, namely-præ, before, and venio, I come or go. Now, you may go before & 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. 5391-357 + 72:33 + 187·21 + 4·2065 + 217·8496 + 42·176 + 52 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. *162 + 134-09 + 2·93 + 97·25 + 3·769230 + 99·083 + 15 + 314. 2, to stop, etc. And how the moral and spiritual imports come |