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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 imparisyllabics with consonantal stems, e is the usual ablative termination, but i 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, alăcer, m., alăcris, f., alăcre, 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 m., f., and n. (accusative, ferocem), but in the plural has for the neuter a separate form in ia, as ferocia.

DECLENSION OF AN ADJECTIVE OF THREE TERMINATIONS.

EXAMPLE.-Acer, acris, acre, sharp, acute, pungent, energetic.
Plural.

Singular.

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Cases. M.

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N.

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G.

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D.

F. acres. acres. acrium. acrium. acribus. acribus.

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N. acria. acrium. acribus. acria. acres.

Ac. acres. acres.
V. acres.
Ab. acribus. acribus. acribus.

acres.

DECLENSION OF AN ADJECTIVE OF TWO TERMINATIONS.

EXAMPLE.-Suavis, m. and f.; suave, n., sweet.

suavium.

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Cases.

Plural.
M. and F.

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suavibus.

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N.
suavia.
suavium.
suavibus.

suavia.
suavia.
suavibus.

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6. The hinge is moved. 7. The becomingness of order delights
mothers. 8. There is a great dust of the ashes. 9. Peacocks are on
10. We have not songs.
the shore.
11. There is a wound in (his)
breast. 12. The light of the region is great. 13. He has a great
name. 14. Pledges are not praised.

EXERCISE 28.-ENGLISH-LATIN.

1. Timesne carbonem ? 2. Cur puerum ferit mater? 3. Decus non est illis. 4. Vulnus est tibi. 5. Tuis patribus sunt vulnera. 6. Vulnera terrent matres. 7. In regione florent poemata. 8. Tibi est 9. Mihi non est pignus. 10. Illis est occasio. Viro magna est occasio.

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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 endless 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 each 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 geometrical 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 perspective 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 guidance 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. audacia, n., bold. 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 use 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

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 anser? 3. Illis sunt anseres. 4. Estne tibi agger? 5. Fulguris odor in pulvinari est. 6. Vectigalia non diligo. 7. Molesti sunt rumores. 8. Pulvinar est ne illis ? 9. Non est illis anser. 10. Tibi sunt pater, frater, et mater ? 11. Illis would make the practice simpler and the result certain. The sunt dolores. 12. Tibi est magnum pulvinar. EXERCISE 27.-LATIN-ENGLISH.

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.

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.

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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 V P1 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

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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 a2 and c2 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., a b2. Perhaps some

add any more lines to that aiready 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 PC 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 P1; 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

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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 a c 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., a 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 practise 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

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 determined, that an analogous method would apply to any circulating decimal whatsoever.

Hence we get the following

bemuntal line or inngut of the eye, BP base of picture. q be the port where the line commences, and from which it retires; as, to mingly the matter, let Ps also be the v P. (The pipi, wà reformser that mi retiring lines vanishing at the point of wight, are lines going off at a right angle with our powition, or with the picture plane. We advise him to turn to page 72 and read the perspective rules and axioms again.) Make the distance from På to Dequal to PS SP. Draw a line Rule for reducing a Circulating Decimal to a Vulgar Fraction. from of to Ps, and on BP make the distance a2 2 equal to the Subtract the number formed by the figures of the non-recurgiven line ab; draw a line from 2 to D, which will cut off the ring part from the number formed by the figures taken to the *pace ake; ase is then the perspective length of ab. The end of the first period, and set down this difference as a numelengths of the retiring sides of planes are determined by the rator. Take as many nines as there are figures in the period, watne ru's, Let it be required to draw a series of retiring and, annexing to them as many ciphers as there are figures in equare winbe Oig. 68). On the base of the picture BP, beginning the non-recurring part, set down the number so formed as a at a, set off any repaired number of divisions to represent the denominator. length of the wide of each slab; from these points, a, b, c, etc., I draw lines to Pa. Find the distance point, D, as in the last case; draw lines from b, c, d, etc., to D, cutting a Ps in ghi. From g, k, i draw lines parallel to the base of the picture, which will complete the squares required; for as ab of the first square i parallel with our position, and touching the picture plane, its true length is therefore shown, whilst ag is its retiring or perspective length.

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 = 8%, 053 = {

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

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. 60 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.

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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 fm 34:567567567.....

Now the difference of 100000 ƒ and 100 f―i.e., 99900 ƒ—must be equal to the difference of the decimals to which they are respectively equal. Now this difference is 34567 - 34, because the infinite recurrence of the period 567 on the right of the decimal point is the same in each decimal, and therefore vanishes when the subtraction is performed.

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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.

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Taking the decimal part separately, ·1415 1415-14-140 Hence 3.1415 31101 = expressed as an improper fraction. Or it may be expressed as an improper fraction at once:3.1415 = 21115-814 = 81101 The truth of this latter method may be established exactly in the same way as the two cases we have already explained.

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9900.

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:-

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13. 052100. 14. 181-082416. 15. 00005-19. 4. ·523. 16. 6125·12527. 30. Approximation. Decimals correct to a given number of places, etc.

We have already remarked, that if we take only a limited number of the figures of a decimal, we approach nearer and nearer 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

decimal, we have
Now taking successively one, two, three, etc., figures of the
is less than by
which is less than

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'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.

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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:

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1. 3.

6. 72.

2. 6. 3. 18.

7. .09.

11. 16.
12. '8567923.

16. .583.

8. '045.

17. 0227.

13. 138.

18. 4745.

19. 5925.

4. 123. 9. 142857.

5. 297. 10. 076923.

14. 53.

15. .5925.

20. 008497133.

2. Change the following sets of decimals to similar and conterminous periods :

1. 6-814, 3-26, and '083.

3. 27, 3, and '045.

2. 46-162, 5.26, 73-123, ·486, and 12·5. 4. 4-321, 6-4263, and 6.

3. Add together the following sets of decimals:

1. 24-132 + 2-23 + 85.24 + 67·6.

2. 328-126 +81-23 + 5·624 + 61·6.

3. 31-62 +7-821 + 8:392 + ·027.

4. 462:34 + 60·82 + 71·164 + ·95.

5, 60-25 + ·31 + 6·435 + ·45 + 45·24.

6. 9814 + 1.5 + 87:26+0.83 + 124.09.

7. 3-6 + 78-3476 + 735·3 + 375 + ·27 + 187·4.

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ENGLISH.-I X.

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 mean-
ing 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 modifica-
tions 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.
exhibit the diversity, and then explain it.
Prevent, signifying to guide, aid forward :-

"Prevent us, O Lord, by thy grace.'

First I will

"-" 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. 5391-357 + 72:38 + 187·21 + 4·2965 + 217·8496 + 42·176 + ·523 to guide or to hinder. The proper meaning, then, of prevent is,

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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 + 184·09 + 2·93 + 97·28 + 3·769230 + 99·083 + 1·5 + ·814. 2, to stop, etc. And how the moral and spiritual imports come

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