persistent under the form of double membranous triangular below, and thus forming an elongated tube ; stamens numerous, leaf-like appendages (Fig. 223).

multi-serial, inserted upon the base of the corolla; ovary inferior, Many species of this natural order are cultivated for the beauty unilocular; placentæ parietal, multi-ovular; berry pulpy ; seeds of their flowers ; some, too, are useful. The succulent leaves numerous, dicotyledonous ; embryo straight or curved; albumen. contain many salts, especially oxalate of lime; some are sapid absent or scarcely visible (Fig. 219). and saccharine. The ice-plant (Mesembryanthemum crystal The Cactacea are American plants; they are ligneous and linum), which is a very common growth in the Canary Islands fleshy; their stem is branched or simple by the suppression of and the Mediterranean region, is charged with gelatinous buds; cylindrical, fluted, flat, or globular, covered with teat-like



vesicles, which canses it to appear, when shining in the sun's | tubercles, the representatives of abortive branches. The leaves rays, as if covered with a crest of hoar-frost. The inhabitants are generally absent, or at most indicated by a small cushion-like of the Canary Islands burn this plant for the purpose of ex excrescence lying beneath a bud; sometimes perfect, plane and tracting soda from its ashes. The fruit of the Hottentot fig. petiolate, as in the case of the Pereskia, or Barbadoes goosemarigold (Mesembryanthemum edule) is eaten as food by the berry; the buds situated upon the axillæ of the abortive leaves Hottentots. Mesembryanthemum fulgidum is a favourite object are of two orders, the inferior ones are covered with spines, of culture on account of the extreme beauty of its deep purple whilst the superior ones are developed in branches or in flowers (Fig. 222).


The berries of many of the Cactus tribe are employed in medi.

cine as a remedy for bilious affections. The Opuntia vulgaris, Characteristics: Calyx adherent to the ovary; with pluri-serial, or prickly pear, has long been naturalised in the Mediterranean petaloid limb, alınost confounded with the corolla ; petals nume- regions ; also the Nopal plant, or Opuntia cochinellifera. Upon rous, pluri-serial, imbricated in æstivation, inserted upon the these plants thrive the valuable cochincal insect, from which summit of a calycinal tube, sometimes free, sometimes coherent carmine and carmine lake are extracted.


LESSONS IN GERMAN.--XXXIX. 15. Dieses Buch hatte einen starten Abgang. 16. Der junge Kaufmann SECTION LXXVIII.-VARIOUS IDIOMATIC PHRASES

erzählte mir, daß der Abgang bercutend zugenommen habe. 17. Je nach

dem e$ mir in den Sinn fomint, reise ich von hier ab. 18. Je nachdem (continued).

er gelaunt ist, fann er der leiblichste, aber auch der unverträglichste Mensch Der Schlag, “the blow, the stroke” (commonly connected with sein. 19. Je nachdem er eg anfängt, wird der Erfolg sein. 20. In so fern rühren), often denotes palsy, apoplexy; as :-Er ist von dem Sdlag ich Dir nüßlich sein fann, will ich es von Herzen gern thun. 21. Er wito gerührt worden, be has been struck with the palsy. Er hatte einen mich mit seinem Rath unterstüßen, in so fern es ihm möglich ist. 22. Sein Anfall vom Schlage, he had an apoplectic fit.

Vater versprach mir die Sache zu befördern, in so fern es in seiner Macit läge. 1. Abgehen=to go away, to leave; as :-Der Zug ist schon abges 23. So etwas ist mir nie eingefallen. 24. Das Concert geht um halb sieben an. gangen, the train has already left (started). Es geht gut ab=it 25. Mein Freund hatte einen herrlichen Einfall. 26. && ist bei dem fröhs sells well; as :-Der Wein geht gut ab, the wine sells well (goes off lichen Deutschen ein Einfal schörer als der andere. 27. Auf die Frage, well).

was ein Einfall wäre, antwortete Einer : , wenn ein Haus einfällt.“ 2. Gr läßt sich nichts abgehen=he lets nothing (advantageous) go

EXERCISE 151. from him, that is, he stints himself in nothing.

3. Je nachdem=ever after, or according as; as:-Je nachdem ich 1. My sister has a cold ; she took cold one wet evening. 2. Muße habe, werde ich Sie besuchen, as, or according as I have leisure That case does not concern me, and therefore I shall not trouble I will visit yon, etc.

myself about it. 3. Has the train already left ? 4. No, it has 4. Einfallen signifies literally, to fall in, or into; hence, to fall not left yet. 5. Has the train left for Oxford ? 6. Two trains down, or to ruin, to decay, etc. With the dative it signifies, to have already left this morning for Oxford. 7. Did the debate come into the mind, to occur; as :-Es ist mir nie eingefallen, fo pass off quietly? 8. No, it was a very stormy one. 9. English etwas zu thun, it never occurred to me to do such a thing.

goods sell well in every country. 10. This grammar has a great 5. So fern, or in so fern=in so far as, if, in case; as :-30 sale. 11. According to your knowledge you will be rewarded. erlaube es vir, in so fern es von mir abhängt, I will permit it, so far as 12. Since he has been struck with the palsy, he has not been able it depends upon me. In so fern es tie Zeit erlaubt, if, or in case the to attend to his business. 13. He was struck with the palsy time permit, etc.

during our visit to your house. 14. As far as it concerns me, I 6. Angehen, used intransitively, signifies, to begin; as :-Der shall take every precaution. 15. In spite of their poverty, Gottesdienst in Deutsøland geht gewöhnlich ted Morgens um neun Uhr an, these people stint themselves in nothing. 16. To mankind the church-service in Germany generally conimences in the morn- nothing is better than a good education. 17. I do not know ing at nine o'clock. Used transitively, it signifies, “ to concern, whether he will grant my request. to be of consequence ;" as :- Das geht ihn an, that is his concern, SECTION LXXIX.-VARIOUS IDIOMATIC PHRASES or that concerns him. Das geht mich nichts an, that does not con

(continued). cern me (is of no consequence to me).

The obsolete word lei (sort, kind) still remains in combination VOCABULARY.

with the numerals, forming what are called the variatives; thus, Abgang, m. sale, mar- In so fern. (See 5, Sinn,


Ginerlei, of one kind, the same; Dreierlei, of three kinds ($ 48); ket ("run"). above.)

as :-Dreierlei bringe ich zu dir, erwähle dir eines, three (sorts of) things

sense. Ab'fühlen, to cool. Kümmern, to concorn, Sigung, f. session, I bring (to) thee, choose thee one. Go ist ihm einerlei or cine, ob An'gehen. (See 6, trouble.


er geht, ober bleibt, it is the same to him whether he goes or stays. above.) Leiblich, tolerable, sup- Un'verträglich, unsoci

1. Gehen, with the preposition über, is often used with the sig.

nification “to transcend, to surpass; Debat'te, f. debate. portable. able, intolerant.

as :-Zufriedenheit geht über Ginfall, idea, Nap, wet.

Vorsicht, f. precau.

Reichthum, contentment surpasses wealth. thought. Rasch, quick, swift. tion.

VOCABULARY. Gin'fallen, to fall in, Schlag, (See Zuʻnehmen, to increase. Begün'stigen, to favour | Gymna'sium, n. gym-, Stüc, n. piece. occur.



Zusammenfallen, to Bonn, R. Bonn. nasium, classical Teich, m. pond. Erkal'ten, to take cold. Schnupfen, m. cold (in tumble, to fall to- Einersei', of one kind, school.

Trägheit, f. idleness. Gelaunt', disposed, the head).

gether, to fall to

the same.

Heilsam, beneficial. Umgebung, f. neighhumoured.

Erzie'hung, f. bringing Lügner, m. liar.

bourhood, environs RÉSUMÉ OF EXAMPLES.

up, education.

Nachtheil, m. disad. Un befümmert, unconDer Saylag rührte ihn auf ter linken The palsy struck him on the Ente, f. duck.


cerned, careless. Seite,

left side.


to re- Nuslos, useless. Universitāt', f. univer&r stand da wie vom Schlag ge He stood there as if struck with


Men'schengeschlecht', n. sity. rührt'.

the palsy.
Fünkchen, n. sparklet. mankind.

Verhältniß, n. rela-
Wo ging der Streit an?
Where did the contest begin ?

Gang, m. direction, Pfeifchen, n. littlepipe. tion,circumstance, Was gehn mich reine Freuden an? How do thy pleasures concern

Rindfleisch, n. beef. situation. (Göthe).

Gerul'dig, patiently. Scaß, m. treasure. Wildpret, n. venison. Das Dampfschiff geht um vier ihr The steam-boat leaves at four Befühl“, n. touch. Schlafen, to sleep. Wohlfahrt, f. welfare. ab.


Geschäft'. n. affair, Schul'tigkeit, f. obli. Zubringen, to pass, Diese Waare geht gut ab. This ware sells well. (See 1.)


gation, duty. spend. Dieser Mann laßt sich nichts abgehen. This man does not stint him.

RÉSUMÉ OF EXAMPLES. self. (See 2.)

Wie es dem Vogel nicht einerlei' ist, As it is not the same to the bird, Die Unterredung ging rubig ab. The conference passed off

ob er sich in dem Ka'fige, oder in whether it is (finds itself) in quietly.

ter freien Luft b.fin'tet, so darf 18 the cage or in the open air, Ie nachdem die Unterhal'tung ist , ist According as the entertainment

cinem Bolte auch nicht eins sein, . so likewise can it not be the auch die Stimmung. is, so also is the humour.

ob es in Sclaverei', oder in Frei same to a nation, whether it In so fern Du Recht hast, werde ich As far as you are right, I will

heit ist.

is in slavery or in freedom. Dir nachgeben.

yield to you.
Dies geht mir über Alles.

This with me excels everything.

Dem Auf'richtigen geht nichts über To the upright nothing is better 1. Mein kleiner Vruter hat ten Schnupfen; er hat fich auf dem Eise die Wahrheit.

than the truth. start erkältet. 2. Wer erhikt ist und sich fu raid abfühlt, fann fich leicht Manchen Menschen geht nicht über With many persons, nothing erkalten. 3. Wir sollen und nicht um Dinge fümmern, welche uns nichts Bequem'lichkeit und Rube.

goes beyond convenience and angehen. 4. In so weit mich diese Sache angeht, habe ich die nöthigen

repose. Schritte gethan. 5. Dieses geht Euch nichts an. 6. Bei vieser Kunde Wir gingen über Mosfau nach Pe. We went by way of Moscow to stand er wie vom Schlag gerührt. 7. Den alten Mann hat der Schlag tersburg

Petersburg. gerührt. 8. Der Mann ist vom Schlage gerührt worden. 9. Wie vom Der Feind ging bei Wien über die The enemy went over the Schlag gerührt sant sie nieder. 10. Diese Waare geht gut ab. 11. Donau.

Danube at Vienna. Wann geht das nächste Dampfschiff ab? 12. Ich sehe nicht, daß fich Es ist unrecht, tie Zeit seines Lebens It is wrong to pass one's life in dieser Mann etwas ab eben läßt. 13. Ist die Sißung ruhig abgegangen? in Ab'gesdiedenheit von den übris seclusion from the rest of 14. Nein, fie ist nitt ruhig abgegangen—bie Debatte war sehr stürmisch. gen Menschen zuzubringen. mankind.


me ?



68 widerfahrt' Manchem mehr Ehre. There happens to many a one 1. @r bringt feine Zeit mit Nichtsthun zu ($ 93. 2). 2. Er brachte

als er verdient'.

more honour than he deben größten Theil seiner Jugend auf den Gymnasien (S 19) und IIniversis täten seines Landes zu. 3. Die meiste Zeit bringt er mit nublofen Bc. Der Vogel ist zum Fenster hinaus'. The bird has flown out of the id aftigungen zu. 4. Biele Menschen bringen ihre Zeit mit Essen, Trinfen geflo'gen.

window. unt Sdlafen zu. 5. Ginem jeden Menschen, der nur ein Fünfchen Gefühl Die Freunde entzwei'ten lich. The friends quarrelled (sepabat, geht nichts über sein Vaterland und über die Wohlfahrt desselben.

rated themselves). 6. Es geht nichts über die Ruhe der Seele, und das Bewußtsein, seine Die Pflaume ist ein Steinobst. (The) plums are a stone fruit. Stuftigkeit gethan zu haben. 7. Er sagte, feine größte Freude und sein Sie verlie'ßen sich darauf, daß er sein They relied upon his keeping grifter Schaß seien feine Kinder, und nichts gehe ihm über dieselben. 8. Verspredy'en halten würde.

his promise. Gin Matrose sagte, es gebe ihm nicht über ein Pfeifchen. 9. Dem Man soll nie eher in eine Sache One should never assent to a Gleichgültigen ift zwar Vieles einerlei ; wer aber sagt, es sei ihm Ades ein'willigen, als biß man diesel'be thing before one has well coneinerlet, ist ein fügner. 10. Was man versprochen hat, soll man halten, wohl überlegt' hat.

sidered it (the same). einerlei, ob Nachtheil oder Vortheil daraus entsteht. 11. Dem Soldaten Ist es nicht, als ob dieses Volf mich Is it not as though this people muk im Kriege Alles eins sein. 12. Ein rechter Mann schickt sich geduldig zum Gotte mache? (Schiller.) would make me a God ? in alle Berhältnisse; e$ ift ihm Alles eins, was er thut, nicht aber, wie

EXERCISE 154. es thut. 13. Seit dem Lobe seiner Kinder ift ihm Alles eins; er

1. Dieses Jahr ist das Obst, sowie alle Früchte, wohl gerathen. 2. ist gleichgültig gegen seine Umgebung, und unbekümmert um den Gang Dieser Baum trägt jedes Jahr sehr viele Früchte. 3. Sind alle Früchte Obft? seiner Geschäfte. 14. Gin jeder Mensch hat seinen freien Willen; teh. 4. Nein, nicht alle, sondern nur solche, bie (s 65.2) an Bäumen wachsen. balb geht es mich nichts an, wie er seine Zeit verwendet. 15. Id reiste 5. Dieser junge Mann verläßt sich zu viel auf seine Verwandten und zu iber Rotterdam und lonton nach Amerita. 16. Der Freund ging soeben wenig auf seine eigenen Fähigkeiten. 6. Er verläßt sich darauf, daß wir ihn über die Straße. 17. Der arme Knabe tauerte ihn, deßhalb nahm er ihn die nächste Woche besuchen. 7. Er verließ sich darauf, daß ihm Gott helfen zu fich in sein Haus, und ließ ihm eine ordentliche Erziehung geben. 18. werbe. 8. Wer sich zu viel auf Andere verläßt, fann leicht getäuscht werden. Ben bas Vieh nicht dauert, und wer unbarmherzig gegen dafselbe ist, den 9. Ich halte (Sect. LXVIII. 2) viel auf meine Freunde. 10. &r hält bauert auch ein Mensch niot.

viel auf ein gemächliches Leben. 11. Dieser Mann halt zu viel von fich EXERCISE 153.

und seiner Klugheit, weshalb er den Rath wohlmeinender Freunde verschmäht. 1. Many people pass their time in idleness. 2. He spent the 12. Nur unter dieser Bedingung kann ich bareinwilligen. 13. Ich willige greatest part of his life in foreign countries. 3. Any man who darein, in so fern (Sect. LXXVIII. 5) es keine üblen Folgen hat. 14. has a touch of honour, renounces no duties which will benefit Er willigte darein, ohne mit allen Schwierigkeiten bekannt zu sein. 15. mankind. 4. He says his greatest treasure was God, and the Dieses Kind thut gerade, als ob es hier zu Hause wäre. 16. Der Matrose whole world is as nothing compared to Him. 5. This man said, stellte sich, als ob er von Sinnen wäre. 17. &r geberbet fich, als ob ihm it were all the same to him whether his undertakings were suc das größte Unrecht widerfahren wäre. 18. Dieser Mann stellt sich, als ob cessful or not. 6. How many sorts of wine have you ? 7. I er beleidigt wäre. 19. Er stellt sich wie ein Kind von fünf Jahren. 20. have three sorts, you may choose which you like. 8. I go every Der Nachbar warf den Zubringlichen zur Thüre hinaus. 21. Der Knabe day twice over London Bridge. 9. Many go to Germany by way ! eilte zur Thüre hinaus, als ich dieselbe öffnete. 22. Zur Thüre hinaus, mer of Ostend. 10. I shall probably spend one month in Bonn. ii. fid entzweit! (Göthe.) 23. &s hängt ganz von den Umständen ab, ob ich My neighbour has three different kinds of ducks in his pond; schon nächstes Jahr nach Amerika reise oder nicht. 24. 68 hängt sehr von they are very beautiful. 12. We have three sorts of roses grow. den Umständen ab, was er thun wicb. 25. Ein so abhängiges Leben die ing in our garden. 13. When I am hungry, it is the same to Bauern in Deutschland Führen, ein eben so unabhängiges führen sie in Ameme whether I have venison or a piece of beef before me. 14. He rika. 26. Ganz unabhängig vermag fein Mensch auf Erden zu werden, bought ribbons of three sorts of colours.

EXERCISE 155. SECTION LXXX.–VARIOUS IDIOMATIC PHRASES 1. Last year the fruit did not turn out well. 2. This tree (continued).

yield fruits but seldom. 3. This young gentleman relies too Berlaffen, when used reflexively, signifies, " to depend upon, to much upon his abilities. 4. No, he does not rely too much rely upon;" as:-Joh verlasse mich auf Ihr Wort, I depend upon your upon his abilities, but he knows it is not well to be dependent word (I leave myself upon your word).

upon those of others. 5. I rely upon you that you will visit me 1. Abhängen, likewise, signifies, “to depend upon, to be depen- next week. 6. Do exactly as if you were at home. _7. The

as :-s hängt von Umständen ab, it depends upon criminal acted as if he were out of his senses. 8. This man circumstances. Thence is derived the adjective abhängig, depen-acts exactly as a child. 9. Where is your canary-bird ? It is dent; as:–Er führt ein abhängiges Leben, he leads a dependent life. flown out of the window. 10. How can I assent to a thing

Die Vereinigten Staaten erklärten sich ale ein unabhängiges Volf, the which is against my inclination ? 11. Whoever quarrels shali United States declared themselves (as) an independent people. be expelled the house. 12. It depends upon circumstances

whether I shall go to my friends. 13. Every man strives to VOCABULARY.

be independent. 14. Depend upon it that I shall not help you Abhängen. (See 1, Frucht, f. fruit. Rana'rienvogel, m. ca- again. above.)

Geber'den, to behave. nary-bird. Abhängig, dependent. Gemächlich, comfort- Deffnen, to open.

KEY TO EXERCISES IN LESSONS IN GERMAN. Betingung, f. condi able, easy,

Umstand, mt. circum

EXERCISE 104 (Vol. II., page 155). tion, stipulation. Gera've, exactly.

stance. Darein'willigen. to con- Gera'then, to turn out, Un'abhängig, indepen

1. For this reason I left my fatherland. 2. He had saved nothing sent. succeed.

3. Several soldiers lost their lives in the battle. dent.

except his bare life. Entgwei'en, to fall out, Hinaus-,out,out there. Verschmä'hen, to dis- sat the preacher. 6. The confederates came together in the night

4. Our troops advanced towards the enemy. 5. Opposite the friend disunite, quarrel. Hinaus'eilen, to hasten dain, despise.

upon the Rutli, conformably to agreement. 7. After the fall of Car. Fahigkeit, f. ability.


Wohl'meinen, to mean thage, the Roman empire hastened more and more towards its dissofelge, f. sequel, con- Hinaus'werfen, to throw well, wish well. lution. 8. Next to the general comes the colonel. 9. I have invited sequence.

| Zu'rringlich,obtrusive. Mr. N. with his children to dinner. 10. We shall visit you, with our

friend, the first opportunity. 11. The Indian goes a-hunting with bow RÉSUMÉ OF EXAMPLES.

and arrow. 12. There have not been similar scenes since the Thirty 36 fann nicht tarein'willigen. I cannot agree to it.

Years' War. 13. You may ask everything of me. 14. Looking to fr wil ligte unverzügʻlich darein'. He agreed (consented) to it un- wards heaven, the sick man expired. 15. Next to him stood the king.

16. Against the wish of his father he entered the army.

hesitatingly, Diese Leute stellen sich, als ob sie von These people act (place them

EXERCISE 105 (Vol. II., page 155). Sinnen wären.

selves) as if they were out of 1. Ich fahre fort, gemäß meiner frühern Bewohnheit. 2. Mein Freund their senses.

ging meinem Feinde entgegen. 3. Mir gegenüber iaf meine Mutter, neben Es wiberfäfrt' uns in unserm Leben There happens to us in our lives meinem Dheim. 4. Ich ritt durch den Parf. 5. Er frägt nach meiner

($ 15. 2. d.) manches Glüc und (many a) much happiness and Schwester. 6. Seit ich dort war, babe ich nichts mehr von der Sache gemanches Unglüd many a misfortune.

hört. 7. Ich habe ihn seit gestern nicht gesehen. 8. 3d habe den Brief

dent upon;

nach seinem Hause gesdridt. 9. Er ging aus dem Zimmer. 10. Ich war b c, to the pp in the points E F G; determine the vp for these bei meinem Bruder. 11. Dies ist gegen das Gesef

lines only, and follow the instructions given with reference to EXERCISE 106 (Vol. II., page 156).

Figs. 27, 28, and 29 (Vol. III., page 9) in drawing the perspective 1. We cannot get through this forest. 2. I received these letters

of the base; the points of contact E and c will be the points to this morning through a kind acquaintance. 3. Along the river they be brought down to e' and d' for the base. The lines of contact saw the enemy's glittering arms. 4. One should be ready to sacri- from F and G must also be brought down to the base of the fice everything for a friend. 5. For this behaviour the father picture upon which to measure the height of the pyramid F'M and punished the boy. 6. They directed the cannons against the town. 7. G'n. Divide f'n or G'n into three equal parts, and through You are not now nearly so frank towards me as was the case formerly. the second from the base draw the line of sight parallel with 8. The enemy shot all the prisoners without exception. 9. Without the pp. Find the VP, to which draw lines from the points of the father's interposition, the children might have caused a great mis- contact E' and c'; these lines cut by visual rays from D and B in fortune. 10. Without doubt my friend will arrive here to-day, 11. In the plan will decide the extent of the base in h i and k. For the order not to increase the mother's fear still more, he did not tell her top lines must be drawn from m and n to the vp, and cut by the truth in all respects. waged war for seven years with Frederick the Second, king of Prussia, visual rays from the plan of the top, as was done with the base; about the possession of Silesia. 13. For this disease there is no draw the inclined edges mh, n i and o k; this will complete the medicine.


PROBLEM XXII. (Fig. 42).-Supposing an equilateral triangle, GEOMETRICAL PERSPECTIVE.–VII.

having its side 2.5 inches, to be the base of a pyramid 2.5 inches

high, draw a perspective representation of the pyramid. Assume PROBLEM XX. (Fig. 40).-Upon the board of the last question one side of the base to be inclined at an angle of 20° with the (see Vol. III., page 73) describe a circle, the circumference touch picture plane, the nearest edge of the pyramid to be į inch from ing the edges.

the picture plane, and the observer's eye to be 5 inches from the picFig. 39, Problem XIX., must be repeated: then upon f b de- ture plane, and 1.5 inch above the horizontal plane on which the scribe a semicircle, and about the semicircle the parallelogram pyramid stands, and opposite a point 2 inches to the left of the fh gb; from , the centre, draw h, %g; through the points angle of the pyramid nearest the picture plane. (From a Military where these last lines cut the circumference draw lines parallel Examination Paper.) to fh or b g, to i, k; next, through the points i, m, k, draw Draw a line, A x, at an angle of 200 with the PP, determine straight lines parallel to f a'c' or bd; draw the diagonals a' d, the point B inch from the PP, and make A B equal 2.5 inches, do, which intersect the parallel lines dropped from the incline upon which describe an equilateral triangle, the base of the k zi; the points of intersection numbered 1, 2, 3, 4, 5, 6, 7, 8, pyramid. The centre of the triangle must be found by bisecting will be those through which the plan of the inclined circle must two of the angles (or by bisecting two of the sides, because the be drawn.

figure is a regular one, having equal angles and equal sides); the Before proceeding with the perspective delineation of the board intersection of the bisecting lines will be the centre at G, which and circle, as shown in the lower part cf Fig. 40, we must detain is plan of the apex of the pyramid. Produce the line co to the pupil whilst we examine the principles upon which the plan D, and draw from A and B parallel lines to meet the pp in E and of the inclined circle is represented. The explanation given F. From E, D and r draw perpendicular lines to the base of the with Figs. 38 and 39 will be sufficient to clear all difficulties picture BP. Place the station point (SP), and draw the HL with respect to the board only. As the circle is lying on the according to the given distance stated in the question; find the board or inclined plane, the end or profile of which is f b, we VP for DG C, which will also be the yp for the other parallel must ascertain the whereabouts of the points through which the lines drawn from the plan to the PP: visual rays drawn from circle is drawn upon the incline. Let the pupil draw a square A, B, C, and cutting other lines drawn from H I K to the VP, will upon a separate piece of paper, and describe within it a circle, at their intersections give the perspective positions of the several then hold the paper at an angle with the horizon, the inclined angles of the triangle, which must be completed by straight edge being opposite the eye, he will first see how from an in. lines forming these angles. Thus far there is no particular clined line we can represent the whole of a square, as illustrated difference in the rule for drawing the perspective of the base from by Figs. 38 and 39, but in this case we have the addition of a the one given for the last problem and several others gone before; circle within the square, therefore the points through which the but we wish especially to draw the attention of our pupils as to circle is drawn must be brought to the edge of the inclined which of the lines of contact E H, D I, or F K must be the one square represented by the line f 6 (Fig. 40); a semicircle upon which the elevation or height of the pyramid is to be set off. will be sufficient to help us in this, as the opposite portions of It will be easily understood, when we consider that the vertex the circle and the several points through which it passes of the pyramid is over the centre of the base, that the line of contact correspond; therefore the method of construction above given connected with the centre must be the one, viz., D I. Therefore will enable us to produce upon the plan of the board the plan upon D i mark the height of the pyramid, viz., I L; from L draw of the circle also.

a line to the VP, and a visual ray from G cutting this line in I To proceed with the perspective representation, let the pupil will give the position of the vertex of the pyramid. Draw from draw visual rays from all the points in c'd and a'b, to cat m lines to meet the angles at the base, which will complete the the respective sides of the perspective projection of the square; representation required. Suppose the three inclined faces had draw lines between the corresponding points on the opposite not been equal, and that the plan of the vertex had been at y, sides of the perspective square, and also the diagonal lines of the then gd must be drawn parallel to a D, the line of contact square : the points through which the circle is to be drawn by brought down, and from the height measured to l a line drawn to hand will be those which are found to answer to the same in the the vp, and the visual ray from g to cut this line, to find the ground plan.

vertex from which intersection the edges drawn to the angles at PROBLEM XXI. (Fig. 41).--A truncated pyramid has a square the base as before will represent the pyramid. base of 1.5 inch side, the top is of 1 inch side, the height 2:5 inches. Suppose the solid to be a regular tetrahedron, that is, a figure Give a perspective representation of the pyramid resting on a hori- with four equal faces, each face would then be an equilateral zontal plane with the pion of the picture inclined to one of the triangle; the height in this case would have to be found. This edges of the bass at an angle of 150 The line of sight to be of obliges us to have recourse to geometrical or orthographic prothe height of the pyramid.

jection. Upon a little reflection the pupil will see that the disAfter placing the line c D (an edge of the base) at the given tance of the vertex from the ground will be less than the length angle, 15° with the PP, draw the plan according to the instruc- of the edge of the pyramid; first, because a straight line drawn tions given in Problem VI.(Vol. II., page 297). Here is an instance from an angle of the equilateral triangle to the centre of the where the use of one vp only will be absolutely necessary; opposite side is less than the side; and again, it would be further there are two sets of retiring lines, viz., C D and its paralle's, reduced because the triangular face is inclined. Now, how much and c B and its parallels; if we were obliged to determine the height may be less than the edge can be determined by the ve for c D and its parallels, we should find by drawing the following mode of proceeding :-Let A B C (Fig. 43) be the from the station point a parallel to c d that the vp would be plan of the pyramid at the base, and the plan of the vertex. at a very considerable and inconvenient distance out of the Now it is understood that all the faces of this solid are equal, aper; therefore produce

the para'lels to C B, viz., A D, a d and and that they are equilateral triangles. Again, we have the full

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