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GEOMETRICAL PERSPECTIVE.-VI. PROBLEM XV. (Fig. 34).-A square pyramid 4 feet side and 6 feet high, one edge of the base at an angle of 60° with the picture plane, and the nearest angle 1 foot within the picture. The base of the pyramid must be treated as the square in Problem X., Fig. 25, Vol. II., page 361; after which draw the diagonals in the perspective projection, namely, a b and c d; their intersection will determine the perspective of the centre e, from which erect a perpendicular e f; this will be the axis of the pyramid. From DE' draw a line through the centre of the base to the picture plane at g, from which draw the line of contact, and upon it from g set off the given height of the pyramid g h. Draw the retiring line from h to DE1, and at the point where this intersects the axis ef will be found the perspective height of the pyramid, namely, at i. Draw

lines from the angles a, b, c, d

to i, to represent the inclined DE 1 edges of the pyramid; this will complete the problem. If we place the ground plan beyond the picture plane (Fig. 35) we must proceed as follows: Place the plan with

give the perspective height of the pyramid. Complete the inclined edges as in Fig. 34. (Observe, we might have drawn ef not parallel to the side of the square; in that case we should have been compelled to find another vanishing point; therefore to save extra work we draw it parallel to the side of the square, so that we may make use of the VP of that side. Definition 13, Vol. II., page 162.)

PROBLEM XVI.-A cone 4 feet diameter and 6 feet high. This will be done from almost the same directions as the pyramid. Look back to Problem XII., Fig. 31, Vol. III., page 9, where we have the perspective of a circle. Now the base of the cone, being a circle, must be treated in the same way. To draw the elevation, draw a perpendicular line, the line of contact from d or b (Fig. 31); mark off upon this line the given height, and from that point draw a line to the respective VP; thus, if the line of contact is from d, DE2 will be its VP; a perpendicular line drawn from the centre of the circle to cut this vanishing line will be the axis, and the point of intersection will mark the apex, from which draw lines to o and p for the sides of the cone.

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the PP, as required by the question, and produce two of the parallel sides to the PP, to be continued perpendicularly to the base of the picture, from which draw the retiring lines to the VP. Visual rays from each angle of the plan, cutting the retiring lines, will give the positions of the angles of the square. This method of using only one VP has been fully explained in Lesson V.; we trust the pupil will make himself master of it, as we shall have to employ it very frequently. For the elevation, diagonals may be drawn and the subject completed as in Fig. 34; but we will show another way, and draw the diagonals in the ground plan which give the centre e. Draw ef parallel to the side of the square, and the perpendicular line fg, the line of contact. Upon f g mark the given height of the pyramid g h, and from h draw a line to VP; a visual ray drawn from the centre e of the plan, cutting the line from h to the VP, will

PROBLEM XVII. (Fig. 36).-A cylinder 4 feet diameter and 8 feet high stands on its end; the eye is opposite half the height of the cylinder. In working this problem we prefer placing the plan beyond the PP, it being necessary to draw a circle for each end of the cylinder, therefore the same perpendicular lines drawn from the plan will answer for both. It will be seen that when these perpendiculars have reached the base of the picture cther lines are drawn from them to the PS, and the circle is drawn by hand as in Fig. 31, Vol. III., page 9. For the upper circle, a b is drawn horizontally across the perpendiculars according to the height of the cylinder, and the same process with regard to the circle is followed as in the one for the base; lastly, lines c, d, drawn tangential to the outer edges of the circles, will give the sides of the cylinder.

PROBLEM XVIII. (Fig. 37).-To draw the perspective represen

tation of an incline. A rod 5 feet long is inclined to the horizon 40°. The plan of the rod is 50° with the picture plane, the nearest end 1 foot from it. In this case the vanishing point of the plan of the rod must be found, and not that of the rod itself. We intend in a future lesson to show how the vanishing point for an incline may be found without a plan, giving only the dimensions and positions, and the method of using it; but for the present turn back to Problem IV., Fig. 14 (Vol. II., page 297), where the same subject is shown in orthographic projection; the rod is there placed at a given angle with the ground, zy, and perpendiculars are drawn from the extremities between

which the line a b, the plan, is drawn. Now we must first project the rod orthographically in order to determine the plan preparatory to drawing it perspectively. An indefinite line a b must be drawn at an angle of 50° with the picture plane; c is the point where the rod touches the ground, draw ce 5 feet long at an angle of 40° with a b; draw e d perpendicularly to a b; cd will then be the plan of the rod; complete the perspective representation of cd, which will be f g (see Fig. 7, Lesson II., Vol. II, page 225). This last observation refers to the perspective only of the plan; we must now represent the rod in its inclined position. As one end of the rod is on the ground, and the other above it, our attention must be directed to the elevated end, because the lower end is already found in g. It must be evident, on turning once more to Fig. 7, that the line fg is the perspective of the line d c; and since the line dc is the plan of the given line e c, therefore e must be perpendicularly over d.

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to this: how far above d? We answer, the length of de, which must be set off on the line of contact, namely, hi. From i draw a line to the VP, and the point m where the line from i to VP cuts the visual ray from d will determine the position of the upper end of the rod; join m g, which will be the perspective representation of the rod.

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PROBLEM XIX. (Fig. 39).—A square board is inclined to the horizon at an angle of 48°; one edge is horizontal, the plan of the inclined edge of the board is 50° with the picture plane; length of side 6 feet. The scale may be either 4 feet or 2 feet to the inch. (We give the pupil the choice, and

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observe that as his knowledge and confidence increase he should, when he repeats the problems, take other angles and other scales of proportion. This kind of repetition will be of great service to him.) We must first show the orthographic projection of the board (Fig. 38), and then apply it to the perspective projection. When the board is horizontal, or laid upon the ground, the plan will be a square, a b c d; but if we raise the side a c, allowing the edge bd to remain upon the ground, it would then be inclined to the horizon as represented by the line or edge of the board, e f; drop a perpendicular from f, then the plan is projected by a b c d. Observe, if the

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edge of the board were still further elevated, the plan would become narrower, that is, a' c' would approach b d. When the board becomes perpendicular to the ground, then the plan would be a line only (see the observations made upon the circle, Figs. 10 and11,Lesson III., Vol.II.,page297). We will now proceed with the perspective projection of the board as given in the question; it is there stated that the edge of the board is inclined at 50° with the PP; therefore draw an indefinite line de at that angle, make db equal to the length of a side, and at a right angle with de; this is the edge upon which it rests, and is horizontal; draw bh parallel to de; draw bf at an angle of 48° with bh, and make it equal to b d; from f draw perpendicularly to bh the line fa' c', we shall then have in the parallelogram a' cbd the plan of the board at the given inclination. The angle d of the board touches the picture plane, therefore d is a point of contact; also the line b a' is produced to the picture plane at m; du and m o are lines of contact upon each of which the height of the inclination of the board a'ƒ is set off as o p and up; from the points p, p draw lines to the VP. Visual rays cutting these lines will give the upper angles of the board, rs; ut sr will be the perspective view of the board.

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Fig. 39.

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After this, we recommend the pupil to apply P the same directions and angles of inclination in representing an equilateral triangle, making the edge equal to db at 40° with the PP; bf, the inclination, will be equal to a perpendicular from the centre of the base to the opposite angle, placed in the plan half way between c'a'. To find the perpendicular, the triangle must be separately constructed.

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erant quum milites a duce e castris in aciem educti sunt. 10. Metuebamus ne urbs ab hostibus obsidione cincta esset. 11. Deus pie colitor. 12. Leges divinæ ne contemnuntor. 13. Sapientes semper ratione regi student. 14. Pueri probe excolendi sunt. 15. Ne vincitor cupiditatibus. 16. Non eris dives nisi divitiæ a to contempta erunt. 17. Contemnens voluptates, diligèris. 18. Quoad literis honos erit, Græci et Latini scriptores in scholis legentur.

EXERCISE 99.-ENGLISH-LATIN.

1. My mind will be cultivated. 2. My brother's mind has been cultivated. 3. If thy mind is well cultivated, thou wilt be loved. 4. Riches are despised by the wise. 5. Riches will be despised by me. 6. Riches will have been despised by my father. 7. Let riches be despised by thee, my son. 8. He strives (studet) to be governed by reason. 9. The boy must be well cultivated. 10. Let the boy be well cultivated. 11. I have taken care that pleasures should be despised by my children. 12. The Latin writers are read in my school. 13. If thou livest well, thou wilt be loved by good men. 14. I fear riches will (may) not be despised by thee. 15. Many wars have been carried on by the English. 16. The city was burnt by the enemies.

THE FOURTH CONJUGATION.

ACTIVE VOICE.

EXAMPLE.-Audio, 4, I hear.

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esse,

Lecti eritis.

Lecti erunt.

Instances. In this way conjugate in full, agor, agi, actum I am driven; regor, regi, rectum esse, I am ruled; and dividor, dividi, divisum esse, I am divided. By conjugating verbs in full after the models given in our lessons, the self-teacher will gain facility in recognising and determining the different tenses of the different moods of Latin verbs of the four conjugations, both in the active and passive voice, at sight. The vocabularies given in the different lessons will supply the student with abundant examples for practice.

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Audiět.

Plu. Audiomus.

Audietis.

Audient.

Sing. Audivi.

Audi(v)īsti.
Audivit.

Plu. Audivimus.

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PERFECT TENSE.

Audi (v)ĕrim.

Audi(v)isse.

Audi (v)eris. Audi(v)erit. Audi(v)erimus. Audi(v)eritis. Audi (v)erint.

PLUPERFECT TENSE.

Audi (v) issem.

Audi (v)isses.
Audi(v)isset.

Plu. Audi(v) erāmus. Audi(v) issēmus.

Audi(v)eratis.

Audi (v) erant.

Sing. Audi(v)ero.'
Audi(v)eris.
Audi(v)erit.

Plu. Audi(v)erimus.

Audi(v)eritis.
Audi (v)erint.

GERUNDS.

Audi(v)issetis.

Audi (v)issent.

Gen. Audiendi. Dat. Audiendo.

SECOND FUTURE TENSE.

tionship society). Vix, scarcely. Voluptas, atis, f., pleasure, delight.

(E. R.

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EXERCISE 98.-LATIN-ENGLISH.

1. Pater curat ut ego strenue excolar. 2. Curo ut puer bene excolatur. 3. Pater curabat ut puer bene excoleretur. 4. Curæ mihi est ut a te diligar. 5. Conjuratio Catilinæ a Cicerone detecta est. 6. Tria bella atrocissima gesta sunt inter Romanos et Carthaginienses. 7. Labor voluptasque naturali quadam societate inter se juncta sunt. 8. Multæ urbes ab hostibus combusta sunt. 9. Vix hostes conspecti

SUPINES.

1. Auditum.

2. Audītu.

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Obedio, 4, I obey, go- | Prodest, he benefits.

verns the dative (obedieratis, syncopated for obediveratis). Paries, -étis, m., a wall. Placéo, 2, I please.

Specto, 1, I regard (id spectant, have this object). Tenuis, -e, thin. Vestio, 4, I clothe (E. R. vest).

Punio, 4, I punish.
Simulac, as soon as.
Sitio, 4, I thirst (sitie-
runt is a syncopated
form for sitiverunt). I
EXERCISE 100.-LATIN-ENGLISH.

1. Milites per totum diem sitierunt et esurierunt. 2. Natura oculos tenuissimis membranis vestivit. 3. Cur domus vestræ parietes coronis ornavistis et vestivistis? 4. Præceptoribus vestris placueratis, quia semper præceptis eorum obedieratis. 5. Vix milites nostri castra muniverant, quum Cæsar aciem instruxit. 6. Non prius dormiemus quam negotia nostra finierimus. 7. Quum milites castra muniverint, ad pugnam se expedient. 8. Cavete, pueri, ne garriatis. 9. Lacedæmoniorum leges id spectant ut laboribus erudiant juventutem. 10. Nemo dubitabat quin pueros semper custodivisses. 11. Narrate mihi qua consolatione ægrum amici animum leniveritis. 12. Nescio cur puerum puniveritis. 13. Non dubitabam quin præcepta mea memoria custodivissetis. 14. Ne garritōte, filiæ. 15. Venio te rogatum ut mecum ambules. 16, Milites urbem custodire debent. 17. Sapientia est ars videndi. 18. Obediendum est præceptis virtutis. navigandi utilissima est.

EXERCISE 101.-ENGLISH-LATIN.

19. Ars

1. They thirsted. 2. I shall be hungry. 3. Thou will not obey my precepts. 4. The boys chatter. 5. They have not obeyed their father. 6. I know not why they have not obeyed their father. 7. No one doubts that good boys obey their father. 8. He kept my words in memory. 9. I shall take care that thou keepest (mayest keep) my words in memory. 10. They come to fortify the city. 11. The art of writing is useful. 12. They adorn the walls of their house with chaplets. 13. I shall not sleep until (before that) I have (shall have) inished my business. 14. Hast thou finished thy business ? 15. He was punishing the boy when I entered the school.

THE FOURTH CONJUGATION.

PASSIVE VOICE.

EXAMPLE.-Audior, 4, I am heard.

EXERCISE 102.-LATIN-ENGLISH.

1. Pater curat ut filius bene erudiatur. 2. Pater curabat ut filius bene erudiratur. 3. Cives metuant ne castra ab hostibus ante urbem muniantur. 4. Oculi tenuissimis membranis vestiti sunt. 5. Quum rex urbem intrabat, omnium civium domus coronis et floribus vestitæ et ornatæ sunt. 6. Non prius dormiemus quam negotia vestra finita erunt. 7. Simulac castra munita erunt, milites se ad pugnam expedient. 8. Metuebamus nę urbs ab hostibus obsidione cincta esset. 9. Improbi puniuntur. 10. Bonus discipulus literarum cognitione erudiri studet. 11. Urbs, obsidione cincta, multis malis punitur. 12. Vir eruditus non solum sibi sed etiam aliis prodest. 13. Pueri diligenter erudiendi sunt.

EXERCISE 103.-ENGLISH-LATIN.

1. They are guarded. 2. The city is guarded. 3. The city will be guarded. 4. The city has been guarded. 5. I take care that the city is (may be) guarded. 6. No one doubts that the city is well guarded. 7. The citizens ought to guard the city. 8. Why do not the citizens guard the city? 9. I know not why the citizens do not guard the city. 10. I fear the citizens may not guard the city. They have prepared for the fight. 12. The walls of the house have been clothed with flowers.

KEY TO EXERCISES IN LESSONS IN LATIN.-XXVII. EXERCISE 94.-LATIN-ENGLISH.

11.

1. I am exercised. 2. Thou art exercised. 3. He is exercised. 4. I was exercised. 5. Thou wast exercised, 6. He was exercised. 7. I shall be exercised. 8. You will be exercised. 9. He will be exercised. 10. Father takes care that I am well exercised. 11. The ditch is filled up. 13. 12. I take care that you are well exercised. I take care that the boy is well exercised. 14. The father took care that his son was well exercised. 15. I took care that you were well exercised, 16. I took care that your daughter was well exercised. 17. Who knows not how our minds are increased by excellent fruits in the pursuit of learning? 18. We fear that our army will (may) be conquered by the enemies. 19. All the citizens feared that the city would be surrounded with a blockade (blockaded) by the enemies. 20. When we are exercised in letters, our minds are increased by the knowledge of many useful things. 21. When we are frightened

Chef Parts: Audior, auditus sum, audiri. Characteristic letter, I long. by a sudden danger, we ought not forthwith to despair of safety. 22.

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23. 24.

The honour of virtue will be blotted out by no forgetfulness. The boys have been strenuously exercised in the study of letters. We feared that the city had been surrounded by a blockade by the enemies. 25. I fear that the soldiers have been frightened by a sudden danger. 26. Let the boy be strenuously exercised. 27. Be not deterred from the design by the difficulties of things. 28. Good scholars endeavour to be exercised in the study of letters. 29. A boy well educated pleases all. 30. The enemies being terrified remain in the camp. 31. Boys ought to be strenuously exercised.

EXERCISE 95.-ENGLISH-LATIN.

1. Pueri strenue exercentur. 2. Strenue exercentor pueri. 3. Pueri strenue exercendi sunt. 4. Pueri strenue exercebuntur. 5. Strenue exercentur pueri. 6. Pueri strenuo exercebantur. 7. Pueri strenue exerciti sunt. 8. Pueri strenue exerciti erunt. 9. Curo ut pueri strenue exerceantur. 10. Curabam ut pueri strenue exercerentur. 11. Mes sorores strenue exercitæ sunt. 12. Puella strenue exercita erit. 13. Metuo ne urbs obsidione cingatur.

EXERCISE 96.-LATIN-ENGLISH.

1. We have led. 2. Thou hast led. 3. Thou leadest. 4. I was leading. 5. He will lead. 6. He may lead. 7. While I was painting, thou wast writing, and brother was reading. 8. The enemies were forming a line of battle. 9. As long as you live, you will live well. 10. If you cultivate virtue, good men will love you. 11. The enemies formed a line of battle. 12. The enemies will draw up their line of battle. 13. We have written many letters (of the alphabet) to-day. 14. The enemies carried on a most frightful war. 15. Cæsar had drawn up in line of battle. 16. As soon as we have written the letters, we will walk. 17. I take care that I cultivate the minds of boys. 18. I took care that the teacher cultivated the mind of my son. 19. No one doubts that I have always diligently corrected the boy. 20. We fear that the enemies have burnt the city. 21. No one doubts that the enemies will surround the city with a blockade (will blockade the city). 22. Tell us what your parents have written. 23. Let him write. 21. Learn, O boy. 25. Good boys learn willingly. 26. The soldier, bravely defending himself against enemies, is praised. We ought to restrain our desires.

EXERCISE 97.-ENGLISH-LATIN.

27.

1. Urbem defendi. 2. Milites urbem defendebant. 3. Urbem defendent. 4. Urbem defenderunt.. 5. Scribebant. 6. Ille literas scripsit. 7. Nemo dubitat quin tu bonas literas scripturus sis. 8. Cura ut literas scribas. 9. Præceptor curat ut discipuli bonas literas scribant. 10. Hodie literas scripsi. 11. Hostes aciem instruent. 12.

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