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GEOMETRICAL PERSPECTIVE.-VIII. PROBLEM XXIV. (Fig. 45).-Draw the perspective view of a flight of three steps, each 4 feet long, 1 foot wide, and 9 inches high; their front making an angle of 40° with the picture plane. The distance of the eye of the observer from the picture plane is 6 feet, from the plane to the nearest point of the object 1 foot. The height of the eye 4.5 feet. Scale at pleasure.

PP.

From a in the picture plane, draw the line a b, at 40° with From c, 1 foot within the PP, make e b equal to the length of the steps, and c d equal to the width of the three steps divided in e and f. The heights to be marked presently on the line of contact. There will be no difficulty in drawing the rest of the plan. Place the station point SP, draw the base of the picture, and the HL three feet and a half above the base, and find the vanishing point. Bring down visual rays from the ends of the steps at both extremities of the plan. Produce dc to h, and g b to k for points of contact, and bring them down perpendicularly for lines of contact. From the base i on ih mark the heights of the three steps one above the other, and also from m, on m k, numbered on both lines 1, 2, 3, and from each of these divisions draw retiring lines to the VP, which, being cut by the visual rays, will give the respective points upon which to draw the ends of the steps, marked again c, e, f, and d; their fronts and edges extend between the corresponding visual rays drawn from the g b end of the plan.

PROBLEM XXV. (Fig. 46).—A rectangular block of masonry 24 feet long, 20 feet high, and 16 feet broad, is pierced by an arch springing at a height of 10 feet, and of semi-circular form, with a span of 12 feet. Let the point of view be on one side of its centre. Distance within the picture plane 2 feet. Height of eye 8 feet. Station point from the picture plane 26 feet. Scale 5 feet to the inch.

We will first draw the perspective view of the arch when the front is parallel with the picture plane. If the pupil has not a scale of inches divided into fifths, he can easily construct one in this manner :-Draw a line, say 6 inches long, to represent 30 feet, and divide it into three equal parts; divide the first division into ten parts, which will represent single feet, and the main divisions will represent tens of feet. Number it similarly to the scales given in Lesson I., Vol. II., page 161.

Draw the PP, and two feet beyond, and parallel with it, draw the line ab equal to 24 feet; a c 6 feet, and c d 12 feet. Draw a e equal to 16 feet, and complete the rest of the plan as shown in the figure. Place the point ƒ a little to the right of the centre, and draw the line ƒ SP, making 9 SP equal to 26 feet. Draw the line BP (base of the picture) anywhere below the PP, allowing sufficient room for the elevation between the base of the picture and the plan above, also the horizontal line 8 feet from BP. Draw visual rays from a, c, d, b, h, i, and bring them down perpendicularly from the PP. Draw a k perpendicularly to the PP, for the line of contact or measuring line for the heights; mark the PS (point of sight) and draw km from k towards PS, stopping at the VR from a. Draw m n parallel to BP, which will be the perspective front of the base of the building. The visual rays from c and d will determine the width of the arch o p. Make the distance kr for the height equal to 20 feet. Draw rs from r as was done from k, and draw st for the top of the building. At u, ten feet from k, draw u y towards the PS, and also y v w; bisect v w; from as a centre being brought down from 7, draw the semicircle v w; the front of the building will then be completed. For the other end of the arch which spans hi of the plan, draw lines vl, wz, from v and w to PS, meeting the visual rays from h and i in land z; join l and %, and either bisect it, or draw a line from a to PS, which, cutting, will give the centre point from which the interior or further end of the arch must be drawn with a radius from the centre to lor z. For the base of the interior of the archway draw lines from o and p, towards PS, cutting the visual rays from h and i; join these points by a line parallel to BP; this will complete the perspective elevation.

Fig. 47. We will now draw the same subject at an angle with our position. Let the angle of the front of the building be 24° with the PP. The other conditions as before.

Draw a b at an angle of 24° with PP, and complete the plan upon a b, as in the last figure. We will use one VP, as in Figs.

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40, 41, 45, and some others. We trust, after all that has been said upon the method of drawing an object with the use of one VP, the pupil will have no difficulty in first drawing the perspective of the block. The principal difficulty will be with the arch, to draw which we shall have to repeat the same principles which were employed for the circle on the board (Problem XX., Fig. 40, page 140); therefore, in order to get the necessary points through which the arch is to be drawn by hand, we must rabat the semicircle. From c as a centre draw the arc f h g; draw dhe parallel to ƒ g, and the semi-diagonals c d and ce through the points where these last lines intersect the arc: draw lines parallel to fd to meet the front of the plan of the building in i and k; visual rays must be drawn from g, k, c, i, f. From the spring of the arch marked on the line of contact at m, make m n equal to fd; the visual ray from c the centre will produce the points o and p; draw the semi-diagonals p r and ps; where these last lines intersect the visual rays from i and k, will give the points through which the arch to u must be drawn by hand. We have not entered into the other part of the work, as we have no doubt that our pupils will be able to do it from the experience they have gained in the solution of previous problems.

PROBLEM XXV. (Fig. 48).—Give a perspective view of a door-frame, a six-panelled door, partly open, the door-frame being parallel to the plane of the picture, and the line of sight two-thirds of the height of the door. (From the Military Examination Papers.) There are very few conditions given. The door is said to be partly open, therefore it may be placed at any angle at pleasure; the wall and door-frame may be placed at any distance from the PP, but they must be parallel to the PP; the proportions of the door and frame are discretionary. This is one of those problems which are frequently given at public examinations with very few working conditions. It gives us an opportunity for advising all who may at any time have to compete in these examinations to use some definite scale in the construction; it will probably save a great deal of confusion and much uncertainty. There will be much in the drawing of this subject that has occurred before, all of which we shall pass over to avoid unnecessary repetition of former instructions. In the plan it must be observed that the width of the door a b must be made equal to a c the space within the frame. The division of a b for the plans of the stiles and panels must be proportionally divided, and those proportions must be set off on a d. (See Lessons in Geometry, Problem XVI., Vol. I., page 209.) There are three lines of contact; the first is from ab produced to the PP. Upon this line of contact all the perpendicular measurements of the stiles and panels are arranged. The second line of contact is from the back of the door produced to the PP. This is for the purpose of arriving at the perspective thickness of the door; therefore from the bases of these two lines of contact at e retiring lines are drawn to the VP; these retiring lines cutting visual rays drawn from the end of the door a in the plan, will give the perspective thickness of the door. The principal retiring lines are those of the top and bottom of the door, and the horizontal edges of the panels, all drawn from the perpendicular measurements above stated. The third line of contact is g h; f g being made parallel to a b for the sake of the advantage of the same VP; a line drawn from the base of h towards the VP, cutting a VE from f, gives the position of the base of the frame i k. The width of the frame across the top is obtained thus:--o being the height of the opening of the door, a line must be drawn from n to m at an angle of 135° with n o; consequently, after mr is drawn, m n will be found to bisect the right angler mi; therefore, the visual rays from the plan of the frame at c cutting the line m n will produce the points in mn from which to draw the mouldings both horizontally and perpendicularly; Ps will be the VP for the interior edge of the frame, as shown in the line drawn from o. The great advantage of using several lines of contact will be seen when working the details. We allude to this for the purpose of observing that it is advisable to draw these lines of contact from produced lines of the plan all parallel with each other, so that one vanishing point may be used for all; otherwise, if they are not parallel, other vanishing points will have to be found, because every retiring line must have its own vanishing

point.

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The terminations -0ev, -01, and -de form adverbs by being added to nouns, pronouns, and verbs, to signify relations of place; thus Oev denotes from a place (whence); i, at a place (where); and de, to a place (whither): for example, oupavo@ev, from heaven; ovpavodi, in heaven; ovpavovde, to heaven. With pronouns de becomes σe, thus aλλoσe, to some other place; so with exe, there, as EKELσe, thither. In the plural of the substantives in -as, -ode passes into Ce, as Aonvace for Anvaσde; from Aenvai, -wv, the city

Athens.

Adverbs of place terminate in -w, as avw, abovе; катw, below; ew, without; eow, within. There are many adverbs which are obviously cases of nouns or pronouns, as ežamins (so in Latin, derepente), suddenly; Tov, somewhere; dπov, où, where; avTOV, on the spot, exactly here or exactly there; ovdaμov, nowhere: these adverbs are all genitives.

Accusatives are also common, as πρwny, at the dawn; μaкpаν, a long way; Teрay, beyond a place, whence the country along the east side of the Jordan had the name of Peræa, that is, the land beyond; dwpeav, gratis, gratuitously; onμepov, to-day (Lat. hodie); avpiov, to-morrow (Lat. cras).

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you make direct interrogatives. Put & before the π, and you
convert the direct into indirect interrogatives, and indirect
relatives. Prefix 7 instead of π, and then you obtain demon-
stratives; as-
Simple Rel.
y, whither,
ἡνικα, when,

όθεν, whence,

oi, whither,

ότε, when,

οὗ, where,

ws, as,

Direct Interrog. πῃ, whither

Indirect I, and R. Demonstratives.

ὅπῃ,

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τῃ, there, thither.
πηνικα, at what time ? δπηνικα, τηνικα, at that time.
ποθεν, whence ?
ποι, whither

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ὁποθεν, τοθεν, thence.

όποι.

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Of these forms οἱ, τοθεν, and τωs are found only in the poets, and consequently are not to be ordinarily used in prose composition.

EXERCISES FROM THE CLASSICS.-GREEK-ENGLISH.

10.

1. Παν το σκληρον χαλεπως μαλάττεται. 2. Ο ουρανος χαλκους εστι τα εξω. 3. Ελεγεν ὁ Βιας, ατυχή είναι τον ατυχίαν μη φέροντα. 4. Η φιλοσοφια διδασκει, ότι δει μητ' εν ταις ευπραγίαις περιχαρεῖς ὑπαρχειν, μητ' εν ταις οργαις περιπαθεις και θηριώδεις. 5. Πως η αχαριστοι, η αμελείς, η πλεονεκται, η απιστοι, η ακρατείς άνθρωποι δυνανται φιλοι γιγνεσθαι; 6. Ο πλουτος και τα εκτος αγαθά χωρίς αρετης ανωφελή εισι TOLS εχουσι. 7. Τις ορνις ευφωνότερα εστιν αήδονος ; 8. Αἱ δευτεραι πως φροντίδες σοφώτεραι. 9. Δαρείου και Παρυσάτιδος γιγνονται παιδες δυο· πρεσβύτερος μεν Αρταξέρξης, νεωτερος δε Κυρος. Γυναίκες ανδρων φιλοπενθέστεροι εισι. 11. Το αδικειν κακιον εστι του αδικεισθαι. 12. Ο Αγησιλαος περι του μεγάλου βασιλέως ειπεν, Τί γαρ εμου μείζων εκείνος, ει μη και δικαιότερος; 13. Ζηνων όρων τον Θεοφραστον επι τῳ πολλους έχειν μαθητας θαυμαζόμενον, ὁ εκείνου μεν χορος, εφη, μείζων, ο εμος δε συμφωνότερος. 14. Σοφός Σοφοκλης, Ευριπίδης σοφωτερος, ανδρων δε παντων Σωκράτης σοφωτατος. 15. Η μελιττα φυσικώς εν τοις ανθεσι εξανευρίσκει το λειότατον μελι και χρηστικωτατον. 16. Παντες, ά επίστανται, ῥᾳστα τε και τάχιστα και καλλιστα και ήδιστα εργαζονται. 17. Καλως και ανδρείως έκαστα ποιεί. Ω Αστυάγης, καλως, εφη, εποίησας, προείπων. 19. Τους ἁλισ κομένους, ὡς κακως κλέπτοντας τιμωρούνται. 20. Αδηλον εστι είτε βελτιον είτε κακιον εσται. 21. Πῶς καλλιον η ευσεβεστερον τιμψη θεους ; 22. Σωκρατης ιδων μειρακιον πλουσιον και απαιδευ τον, ίδου, εφη, χρυσουν ανδραποδον.

VOCABULARY.

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14. Ανδρ. παντ. σοφωτ. etc. genitive; thus we say in English, "the fairest of women.' 15. Φυσικως, naturally, by natural impulse ; λειος, -α, -ον, soft, mild, sweet; χρηστικός, -η, -οv, useful.

18. Εποίησας (from ποιειν), thou hast done; προειπων, in fore telling (that).

19. Τους άλισκ. the captives; ἁλισκειν, to take, capture; κλέπτειν, το του; τιμωρεομαι, Ι punish.

20. Εσται, it will be, future of είναι.

21. Τιμψη, could he honour ? τιμαω, I honour.

22. Μειράκιον, a young man ; ανδραποδον, -ου, το, a slave.

EXERCISE 61.-ENGLISH-GREEK.

1. Wise men seek not external advantages. 2. Women suffer
very much in adversity. 3. An intemperate man cannot become
a faithful friend. 4. The nightingale is the sweetest (in
voice) of birds. 5. Girls are more given to sorrow than women.
6. The wisest (man) is greatest. 7. I am admired for having
much wealth.
8. How can men admire me for having much
wealth 9. My brother is wise, my father is wiser, the philo-
11. Fight, Ο citizens, well and bravely for (περι) your (the)
sopher is wisest. 10. Children naturally love their parents.
city.

KEY TO EXERCISES IN LESSONS IN GREEK.-XVII.
EXERCISE 57.-GREEK-ENGLISH.

1. The deepest sleep is the sweetest. 2. Many flowers yield a very sweet smell. 3. Nothing is swifter than youth. 4. The incontinent serve a most vile servitude. 5. Friendship is the sweetest of all things. 6. Nothing is more disgraceful than to have one thing in the mind but to say another thing. 7. Serpents are most odious to all other animals. 8. Nothing is more hateful to man than man. 9. Swiftly does time bear things away with it.

EXERCISE 58.-ENGLISH-GREEK.

1. Ουδεν ήδιον εστιν η βαθυς ύπνος. 2. Ήδιστος εστιν ύπνος. 3. Ουδεν αισχιον εστι της δουλειας. 4. Οξύτατον εστι δουλεια. 5. Ιπποι εισι τάχιστοι. 6. Ουδεν έχθιον εστιν η βουλη κακη. 18. 7. Αισχιον εστιν αλλα μεν λεγειν, αλλά δε εν τῳ νῳ ἔχειν. 8. Οἱ πονηροι αλλα μεν εν τῷ νῳ έχειν, αλλά δε λέγειν. 9. Ουδεν εστιν ήδιον πιστου φίλου.

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3. Τον μη φερ., that he who could not bear misfortune.

4. Ευπραγία, -as, ή, prosperity, literally, well-doing, from εν and πραττειν, I do, I am in a certain condition, as in our phrase, "How do you do?" περιχαρής, very joyful, περι gives the idea of much or excess; οργή, ης, ή, anger, here used for adversity, considered as a consequence of the anger of the gods; περιπαθης, -es, sufering greatly, acutely sensible to sufering; θηριώδης, -ες, like the animals, low, downcast.

5. Αχάριστος, -η, -ov, ungrateful; αμελης, -es, neglectful; πλεονεκτης, αvaricious; απιστος, unfaithful.

6. Εντος, adv. without; τα εκτ. αγ. external goods, advan tages; ανωφελης, -es, useless ; χωρις, apart from; τοις εχουσι, το those who have (them), that is, their possessors.

8. Πως, how, somehow, some way, in a measure; the adverb restricts or qualifies the statement.

9. Δαρείου και Παρυς. These genitives depend on παιδες; we should say, D. and P., have two sons.

EXERCISE 59.-GREEK-ENGLISH.

1. The longest life is not the best, but the most virtuous. 2. Moderation is the best in all things. 3. The judgments of the more aged are the better. 4. No counsellor is better than time. 5. Either utter things better than silence or keep silence. 6. That which is most secure is always the best. 7. You jeer, O excellent friend. 8. At times cowards are more fortunate than brave men. 9. There is no worse evil to man than grief. 10. Flattery is the worst of all the other evils. 11. A man soft in soul, and even capable of bribery. 12. Prudence is the fairest virtue for women, 13. There is no nobler possession than a friend. 14. Slavery is most painful to a free man. 15. The way is very long. 16. The crocodile grows very big from being very little. 17. The earth is less than the sun. 18. Be content even with less. 19. Very few men are happy. 20. No law is stronger than necessity. 21. Small gains often produce greater damage. 22. No evil is greater than anarchy. 23. War brings very many evils.

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1. Ουδεν βελτιον πρακτικού βίου. 2. Η δόξα των παλαίων εστι κράτιστα. Ο χρόνος εστι συμβουλος άριστος. 4. Το ασφαλεστατον εστι κρατιστον. 5. Η λυπη εστι κακον μέγιστον. 6. Ουδεν εστι χείριον η κολακεία. 7. Ο ακρατης τους ηδοναις δουλεύει. 8. Γυναιξί εστιν ουδεν καλιον της σοφίας. 9. Τις ελεύθερη ουδεν εστι κακιον της δουλειας. 10. Ο κροκοδειλος μήκιστος εστι. 11. Ο ύπος μείων του πατρός. 12. Τοις κακοις εστι πολλακις πλείονα η τοις αγαθοις, 13, Πόλεμος φερει πολλα κακα. 14. Το μεν κελεύειν εστι ράδιον, το δε πε

10. Φιλοπενθης, -es, fond of mourning; πενθος, -ους, το, grief, θέσθαι ισχυρόν. 15. Οἱ καρποι πεπαιτατοι εισιν ἥδιστοι, 16. Τα πρόβατα

lamentation.

11. Το αδ. The infinitive mood with the article is often equivalent to a noun in English: to injure another is worse than to suffer an injury.

12. Ο μεγαλου βασιλεως, the great king, that is, the king of Persia, who was the great king to the Greeks; εκείνος, he, that one.

13. Όρων, seeing, pres. part. from οραω ; επι τῳ, etc., in consequence of having many disciples; xopos, our chorus, here class, audience, σύμφωνος, -ov, agreeing, harmonious ; ὁ εμος, mine; literally, the mine.

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well with the key-note. They are its dissonances. It is generally confessed by teachers of singing that RAY and TE are the most difficult," the most artificial," notes of the scale. This is especially noticeable, in uncultivated voices, on the note TE in ascending the scale, and on RAY in descending. The mathematical musician who wrote the article "Scale" in the "Penny Cyclopædia" states, that a note sharper than the present RAY and one flatter than TE would be more natural. General Perronet Thompson, however, seems to have come nearer the practical truth in his discovery of what he calls "the duplicity of the dissonances." He believes these notes to be "duple" or double, and asserts that a good ear and voice will intuitively choose that one of the two which best accords with the accompanying notes. RAY, for instance, in any near fellowship with FAH and LAH will be slightly flattened, but to tune well with SOH will remain in its sharpened form. He has constructed instruments upon these principles which have approached more nearly the "just intonation" of the good singer or violinist than anything before attempted. We have heard musicians say, that to hear Mr. Purkis play on General Thompson's organ

in "the Great Exhibition" of 1851 was the most delicious treat imaginable to true and delicate ears. Unfortunately, his keyboard looks discouragingly complex. We shall have to speak of these points more definitely when we come to treat of harmony. We have mentioned the difficulties connected with these notes that the pupil may understand why they are placed last in the order of illustration. It was necessary to make him familiar with the simpler and easier notes of the scale before he advanced to any characteristic or difficult use of these two. Our pupils will also see the mistake of the ordinary methods of developing the voice, by practising it at once on the scale of successive degrees, instead of the chord of easily recognised intervals, to which the more difficult notes may be afterwards added.

2. The note TE, when heard at length, and after the ear has been filled with the other notes of the scale, inspires the mind with a feeling of suspense accompanied with a strong desire for its resolution in the key-note. For this reason, it is generally called the "leading note"-leading the ear to the key-note. To prove this, let the pupil sing the notes of the scale either up or down, and try whether the ear can be satisfied by resting on TE.

3. The note RAY, when heard in similar circumstances, excites a feeling of suspense almost as strong as TE, but does not so decidedly indicate its resting note. The ear is pleased by its rising to the third of the scale (ME), but more satisfied by its falling on the key-note. Try the well-known Gregorian Tone with these two endings:

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Take,

- 1

-:1

t

d' mr.d 0 take me to your

d:

care.

After several repetitions and variations of the softer theme referred to, the whole piece closes with this last piercing appeal. Indeed, examples from the great masters might be multiplied easily, but space forbids us to enlarge.

5. It may easily be noticed that every tune is divisible into parts which correspond with a line in poetry. The notes most frequently used for the close of these phrases are DOH, ME, SOH, the resting notes of the scale. Next to these in frequency are RAY and TE. But the more emotional notes, FAH and LAH, are but seldom found in that position. Let our pupils test this interesting fact by examining the previous exercises. They are now invited to study the following exercise, the words of which were written by Robert Burns, the national poet of Scotland, but are slightly altered here to obviate the difficulty of pronunciation which the Scottish form of many of the words might present to some of the students of our Lessons in Music. A finer lyric, as far as the sentiments of the poetry goes, was, perhaps, never written. Every line has a genuine ring which must find its way to the heart of every man who hears or reads the poem. The theme is a noble one, and its dignity is in no way marred by the simplicity of the language in which it is couched.

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