A compact rock has its particles in close and fiệm proximity, but coverings, mixed with the shells of larger inhabitants of the one which is friable permits of being crumbled. A porous speci- ocean. men is full of holes, like pumice-stone, but when the pores are Marble is also carbonate of lime which has been submitted to small it is said to be cellular or vesicular. Rocks are hard or the action of heat, and thereby rendered crystalline. soft, but beyond a general application these terms are not much Mountain limestone is so called because it appears in ranges of in vogue. In mineralogy we shall find that much stress is laid hills. It is very compact and dark-coloured. The walls in many on the comparative hardness of minerals. parts of Ireland are built of this admirable material, which is Rocks which are composed of water-worn pebbles cemented found in great quantity in that island. In Sweden there is an together are conglomerates. If angular fragments take the instance where a vein of granite has passed through a mass of place of the pebbles, a breccia is the result, which word is the mountain limestone. This it rendered like marble where it had Italian for "fragment.” A pebble is a small piece of mineral contact with it, and the white crystalline rock gradually shaded matter, worn round by the action of water. When the sea-beach away to its original dark condition. The colour of this stone is is not sand, but stones rounded by the waves, the accumulation considered to be due to carbon in a state of minute sub-division; of pebbles is a shingle. but the quantity is so small as to evade detection by chemical If the size of the pebble exceed a few pounds in weight, it means. becomes a boulder. Oolite is a species of limestone which appears in little round A stratum which has been deposited by aqueous agency is of grains, like the roe of fish ; hence its name. Each of these one of three characters. It is either of an arenaceous quality-grains has a particle of sand for a nucleus, round which the that is, a sandstone; or it is argillaceous—that is, a clay; or it limestone seems to have congregated. is a limestone. When magnesia is present in a quantity as much as thirty In each of the great systems in which rocks are classified, five per cent., the rock is termed dolomite. these three species of rocks appear. Each system has its lime When clay and sand are mixed in a considerable quantity with stone, its sandstone, and its argillaceous rocks, or its clay beds. lime, a loam is formed. When only a little sand is present, and Sand, of which arenaceous rocks are composed, is properly the mixture is of a more firm texture than a loam, it is denomismall particles of quartz, which is the hardest constituent of nated a inarl. It is not a usual thing to find any limestone granite, and therefore most successfully resists the process of pure; they all contain an admixture of foreign matter. attrition. These are the general characteristics of the great divisions of Quartz is pure silex or flint, and may at once be recognised in rocks. Those deposits which are peculiar to the various formaa specimen of granite, as those clear, semi-transparent grains. tions will be alluded to in their proper places. When granite decomposes, these grains remain intact, or nearly 80; for silex is most difficult to dissolve. Particles of sand are always rounded, as if by the action of running water. LESSONS IN GREEK.-XXII. Sandstone is an aggregate of these particles, which are held THE NUMERAL ADVERBS together in some cases without any apparent cement; but often calcareous or siliceous matter is present in a small quantity, DENOTE how many times a number is to be taken, as “sis which binds the arenaceous particles together. Sometimes clay times six make thirty-six.” Here six times is a numeral adverb; or the oxide of iron forms this cement, and to this latter sub. thus, dus signifies twice, Tpus three times. The termination of the stance the red colour which frequently tints sandstones is due. numeral adverbs is in general-kis (-akus, -Takes), which is annexed Some fine-grained sandstones are found which contain minute to a cardinal, as terrapakis, égakis, èKUTOVTAKIS. scales of shining matter; this is mica, another constituent of granite, and such rocks are termed micaceous sandstones. These 1. árat. 18. οκτωκαιδεκακις. . sandstones are found among the earlier formations. 2. Ous. 19. EvveaKAIDEKARIS. Grit is applied to sandstones whose grains are very coarse. 20. εικοσακις. . Argillaceous Rocks.-Any earth which is sufficiently plastic to 4. τετρακις. . 30. τριακοντακις. . be kneaded by the hand is, in common language, a clay. Strictly 5. πεντακις. . 40. τετταρακοντακις Or speaking, clay is composed of siliceous and aluminous particles. 6. εξακις. . [τεσσαρακοντακις. Kaolin, or porcelain clay, is the finest of the clays. Its composi 7. επτακις. . 50. TEVTIKOVTAKIS. tion is given in this table, in which it is compared with other 8. OKTARIS. 60. εξηκοντακις. . specimens : 9. εννεακις, εγνακις. 70. έβδομηκοντακις. . 10. DEKUKS. 80. oydonkortaKIS. 90. ενενηκοντακις. . 100. εκατοντακις. . 13. τρισκαιδεκακις. . 200. διακοσιακις. . Silica 64-10 53.66 49.44 14. τετταρεσκαιδεκακις Or τεσ 300. TPIAKOJIAKES. Alumina. [σαρεσκαιδεκακις. 1000. χιλιακις. . Iron Oxide 1.85 1:35 2000. δισχιλιακις. . Lime 0:40 1.20 10,000. uuplakis. 17. επτακαιδεκακις. . 20,000. Souvlakis. Potash 1.9 RECAPITULATORY EXERCISES FROM THE CLASSICS. Water 11.2 10.00 12:08 1. Αναχαρσις κρειττον ελεγεν, ενα φιλον εχειν πολλου αξιον, η πολλους μηδενος αξιους. . 2. Αννων, ο πρεσβυτερος, εκ της 99.9 99.49 100-23 100.00 Λιβυης επερασε μεγαλην δυναμιν εις Σικελιαν, πεζων μυριάδας πεντε, ιππεις δε εξακισχιλιους, ελεφαντας δε εξηκοντα. 3. Τους All days are formed by the very fine disintegrated particles | Σηρας ιστoρoυσι μεχρι τριακοσίων ζην ετων, και τους Χαλδαιους of water-worn rocks. Mud is matter also formed of such par- Tep Ta ekatov ETN Biovy loyos (Tl). 4. Αργανθωνιος, ο ticles, but it contains a mixture of animal or vegetable matter, TapTnOOLWV Baoilevs, #EVTNKOVTA KAL Ékatov ethn B.Wrai deyetdi. or both. Shale is hardened clay, but if it be softened by water 5. 'O MatwV ETENEUTICE TW IPWTW TWS Odons keu ŠKATOOTUS it will exhibit the plastic properties of that body. All clays | Ολυμπιαδος, βιους ετος εν προς τους ογδοηκοντα. 6. Δημητριος when breathed upon emit a peculiar earthy smell. ειπε τω Νερωνι· συ μεν απειλεις εμοι τον θανατον, σοι δε η Calcareous Rocks.-All rocks of this description may be at | φυσις. 7. Σχολαστικος απορων, τα βιβλια αυτου επιπρασκε, και once discovered by the application of a few drops of any acid. | γραφων προς τον πατερα ελεγε, συγχαιρε ημιν, πατερ, ηδη γαρ ήμας Efervescence at once takes place, owing to the escape of carbonic | τα βιβλια τρεφει. 8. Αναχαρσις και Σκυθης ερωτηθεις υπο τινος, acid gas, which is always associated with lime, forming a car τι εστι πολεμιον ανθρωπους και αυτοι, εφη, εαυτοις. 9. Σχολαστικος bonate of that base. οικιαν πωλων, λιθον απ' αυτης εις δειγμα περιεφερε. 10. Κριτης There are many varieties of limestone. Chalk, as we have | ων, αει ταυτα περι των αυτων γιγνωσκε, ουδεν προς χαριν ποιον said, is due to the incessant work of minute animalcules, which 11. Yuxns et luenov Tos ceavtov. 12. Bovaov aperkei #20L . 13. separate the lime from the sea-water to form their calcareous navwV Mariota GEAUTOV ALO Xuvou. 14. 'PqOTOY SHAPTW FOTW THE NUMERAL ADVERBS. Kaolin. Fire Clay. Blue Brick Olay. 50:5 33.7 1.8 23:15 32.00 46.38 34.26 1.48 Soda 100-05 Tis σοι πανεται. αυτον εξαπαταν. 15. Ο αγαθε, μη αγνοει σεαυτον. 16. Ιφικρατης | αναπεσειν (πιπτω, I fall), to sit down ; χορτος, -ου, δ, grass ; σκυτοτομου μεν υιος ην, ενδοξοτατος δε. Ούτος ειπε προς τινα ανεπεσον, they sat down; τον αριθ., as to number, that is, τη των ευγενων" το μεν εμον γενος απ' εμου αρχεται, το δε σον εν | number, Or to the number ; ώσει, αbout. 17. Θαλης ερωτηθεις, τι κοινοτατον; απεκρίνατο 2. Των ακουσαν. (ακουω, I hear), of those who heard; επιστευελπισ" και γαρ οίς αλλο μηδεν, αυτη παρεστιν. 18. Οίον το σαν (πιστιs, faith), believed; εγενηθη (γινομαι, I become), κας, τθος έκαστου τοιουτος ο βιος. 19. Φερεται και Νειλος απο των rose to. Αιθιοπικων ορων μεχρι της εις θαλασσαν εκβολης σταδια μυρια 3. Ειδον (ειδος, αρpearance, shαρε), I saw ; ηκουσα, I heard; και δισχιλια. 20. Τα δις πεντε δεκα εστιν. 21. Εντευθεν εξε- αρνιον, -ου, τo, lamb ; εσφαγoμενον (σφαγιον, α υictim). λαυνει σταθμους δυο, παρασάγγας πεντε, επι τον Σαρον ποταμον, 4. Ο εχων νουν,.let him who has ; ψηφισατω (ψηφας, α ου ην ευρος τρια πλεθρα. pebble ; the Greeks usually reckoned with beans, as the Latins VOCABULARY. did with pebbles, calculi, whence calculate), calculate. 1. Αξιος, -α, -ον, worth, worthy και πολλ. αξ., of great value. 5. Διεκωλυε (κωλυω, I hinder), tried to hinder ; βαπτισθηναι, 2. 'Avvwv, -wvos, d, Hanno, the Carthaginian general. Emepage to be baptised; Bantw, I dip; epx?, comest thou? (from περας, beyond), transported, carried over; πεζων (from 6. Βασταζετε (βασταζω, I carry), bear ; ουτως, thus ; αναπλη. πεζος), of foot-soldiers; ιππεις (ίππευς), horsemen, cavalry. (ανα, up, πληροω, I fill), fill up, fulfi. 3. Σηρας (Σηρ, -ος), the Seres, αη Eastern people who produced 7. Καυχησις, εως, ή, boasting; συνειδησις, εως, ή, conscience; ελλ; ζην (infin. of ζαω, I live), to live ; Χαλδαιους, the chaldeans; “απλοτης, -ητος, ή, simplicity και ειλικρινεια, ας, ή, sincerity; τα εκατον ετη, literally, above the hundred years (so with either σαρκικος (σαρξ, flesh), fleshly ; ανεστραφημεν, we have behαυed (connumber the article is used when a whole is contemplated; in con- ducted) ourselves, we have acted ; TEPLOCOTEPWs (Tepi, denoting struing into English you must drop the article in such cases) : abundance), more exceedingly. βιουν (from βιοω, I live, βιος, le), to live. 8. Παρακλησις, εως, ή, exhortation, comfort ; παραμυθιον, -ου, 4. Βιωσαι, to have lived; λεγεται, is said. τo, solace, soothing και κοινωνια, ας, ή, community και πνευμα, -ατος, 5. Ετελευτησε (from τελος, αη end), came to an end, died; spirit; σπλαγχνον, -ου, το, bowels ; οικτιρμος, -ου, ο, ρίty ; Ολυμπιας, -αδος, ή, αη Olympiad, a period of five years (the πληρωσατε (πληροω, I fill), fulfil ; φρενητε (φρενες, the mind), Greeks reckoned time by Olympiads, as we date from the birth that ye desire, aim at, love; rouluxou (yuxn, the soul), being of of Christ, A.D.); βιους, λαυing lived; ετος εν, etc., one year to the same soul, of one soul και εριθεια, ας, ή, strife; κενοδοξια (κενος, eighty, that is 81 years. empty), υαλnglory και ταπεινοφροσυνη, -ης, ή (ταπεινος, humble), 6. Ειπε, εαid; Νερων, -ωνος, o, the Roman emperor Nero ; louliness of mind; ηγουμενοι, thinking, considering; υπερεχειν, απειλεις (from απειλεω, I threaten), threatenest. to be superior; σκοπουντες (σκοπειν, to look, hence επισκοπειν, to 7. EKONQOTIKOS, -ou, d, an idler, a witling; atopwy, being in overlook, whence our word bishop), looking at. dificulties ; επιπρασκε, sold. REMARKS. 8. Ερωτηθεις (ερωταω, I ask), being asked; εφη, said, answered. 9. Δειγμα, ατος, το, α εpecimen και περιεφερε (περι and φερω), | Consequently, if in two languages the pronouns are found to The pronoups are among the oldest words in every language. carried about. 10. Γιγνωσκε, pronounce the same judgment; προς χαριν ποιων, these two languages are alkin to each other. Such marks of have strong marks of resemblance, we may safely conclude that doing nothing for favour. 12. Αρεσκειν, to please ; βουλου, υish (try). resemblance may be found by comparing the Greek and the 13. Αισχυνου (αισχυνομαι), reverence. English personal pronouns together. Thus the Greek εγω, 14. Εξαπαταν, to deceive, cheat. through the Latin ego, is clearly the English I (also the German 15. Αγνοει, be thou gnorant. ich and the French je). Look at the Greek accusative Me, the 16. Σκυτοτομος, -ου, o, a leather cutter (from σκυτος, -ους, τo, the Latin tu, and the English thou ; also the accusatives, Latin me, and the English me. Again, compare the Greek ou, hide, leather); evyerns, well-born ; apxet. an. euov, literally namely, ce, te, thee. The è (the e aspirated and so made he) is begins from me, that is, with me; Tuvetai, comes to an end. obviously our he. 17. Απεκρινατο (απο and κρινω), αnswered; ελπις, hope ; και γαρ, for. Similar remarks may be made with regard to the numerals. 19. Φερεται, carries itself (middle voice), ποιος και εκβολη, -ης, ή, | Greek numeral system is the same as our own. Obviously in structure, as well as in individual numbers, the a falling out of; uexpu, up to, down to, until. 21. Εξελαυνει, marches. The student, if he has well attended to these lessons, may y now rejoice in having made some considerable progress; and the EXTRACTS FROM THE NEW TESTAMENT. progress he has made he may in a measure estimate by the com1. Ειπε δε ο Ιησους, Ποιησατε τους ανθρωπους αναπεσειν. Ην | parative ease with which he has just read passages from the δε χορτος πολυς εν τω τοπω. Ανεπεσον ουν οι ανδρες τον αριθμον | Greek New Testament. ώσει πεντακισχιλιοι (John vi. 10). 2. Πολλοι δε των ακουσαντων GENERAL VIEW OF WHAT HAS BEEN SET FORTH. τον λογον επιστευσαν και εγενηθη ο αριθμος των ανδρων ώσει χιλιαδες πεντε (Acts iv. 4). 3. Και ειδών και ηκουσα φωνην Noun Substantive, used to name objects, as στρατιωτης, soldier αγγελων πολλων κυκλω του θρονου και των ζωων και των πρεσ (a soldier). βυτερων και ην ο αριθμος αυτων μυριάδες μυριάδων και χιλιαδες | Article, used to determine nouns, as και στρατιωτης, the soldier. χιλιαδων, λεγοντες φωνη μεγαλη, Αξιον εστι το αρνιον το εσφαγο of quality, αγαθος στρατιώτης, good soldier. . μενον λαβειν την δυναμιν και πλουτον και σοφιαν και ισχυν και of number, δεκα στρατιωται, ten soldiers. τιμην και δοξαν και ευλογιαν (Rev. ν. 11, 12). 4. Ο εχων νουν of order, tenth legion. ψηφισατω τον αριθμον του θηριου αριθμος γαρ ανθρωπου εστι, ουτος ο ανθρωπος, this man. και ο αριθμος αυτου (understand εστιν) χξς (Rev. xiii. 18). εκεινος ο ανθρωπος, that man. 5. Ο δε Ιωαννης διεκωλυεν αυτον, λεγων, Εγω χρειαν εχω υπο σου demonstrative, και αυτος ανθρωπος, the same man. βαπτισθηναι, και συ ερχη προς με; (Μatt. iii. 14). 6. Αλληλων αυτος ο ανθρωπος, the man himself. . τα βαρη βασταζετε, και ούτως αναπληρωσατε τον νομον του ανθρωποι τινες, some men. Χριστον (Gal. vi. 2). 7. Η γαρ καυχησις ημων αυτη εστι, το interrogative, τίς ανθρωπος, which man? μαρτυριον της συνειδησεως ήμων, ότι εν απλότητα και ειλικρινεια relative, ο ανθρωπος ος, the man who. Θεου, ουκ εν σοφια σαρκικη αλλ' εν χαριτι Θεου, ανεστραφημεν εν possessive, , ο εμος πατήρ, my father. . τω κοσμο, περισσοτερως δε προς υμας (2 Cor. i. 12.) 8. Ει τις Pronoun, εγω, Ι; συ, thou ; ου, of himself. συν παρακλησις εν Χριστώ, ει τι παραμυθιον αγαπης, ει τις κοινωνια τΡευματος, ει τινα σπλαγχνα και οικτιρμοι, πληρωσατε μου την KEY TO EXERCISES IN LESSONS IN GREEK.-XXI. χαραν, ένα το αυτο φρονητε, την αυτην αγαπην εχοντες, συμψυχοι, το εν φρονούντες, μηδεν κατα εριθειαν η κενοδοξιαν, αλλα τη ταπει EXERCISE 70,-GREEK-ENGLISH. νυφροσυνη αλληλους ηγουμενοι υπερεχοντας εαυτων, μη τα εαυτων 1. The river Euphrates is four stadia in breadth. 2. The stadium έκαστος σκοπουντες, αλλα και τα έτερων έκαστος (Philippians i. of the Romans contains one hundred and twenty-fve steps, or six hundred and twenty-five feet. 3. To Cyrus there were present VOCABULARY. thirty-five ships from Peloponnesus. 4. The breadth of the Sarus, a river of Cilicin, was three plethra. 5. The plethron coutaius a hundred 1. Ιησους, Jesus ; ποιησατε (ποιεω, I male, do), make, cause to; teet. 6. Cydius, a river of Ciliciu, 15 ένο plethra wide. 7. The Adjective οκτω μηνες. 400 breadth of the Meander, a river of Phrygia, is twenty-five feet. 8. The upon the PP diminishes. Turn to Fig. 24, Vol. II., page 360, parasang, that is, Persian measure, contains thirty stadia, or eighteen | where the slabs of the pavement touching the PP are drawn to thousand seven hundred and fifty feet. 9. The calculation of the the size given by the scale; also f e, the perpendicular edge of entire journey, the expedition and retreat, which is described by the cube in Fig. 33, Vol. III., page 9, is another example. After Xenophon, was two hundred and fifteen stadia, one thousand one hundred and fifty-five parasangs, thirty-four thousand six hundred and this remark, it will be seen that the object may be made to touch fifty stadia ; and the length of time of the expedition and retreat was the Pp in more than one place, if it is placed at a distance from a year and three months. 10. The friendship of one intelligent person the pp, by means of one or more of its lines being produced to is better than the friendship of all unintelligent people. 11. This was the PP as points of contact. Therefore, if we have the option of the number of the army of Cyrus : of the Greeks there were ten thou- placing a line representing the PP anywhere in conjunction with sand four hundred hoplites, two thousand five hundred targeteers; but one of these points of contact, besides eur usual practice of of the Barbarians with Cyrus, there were one hundred thousand, and putting it below the drawing, we have the advantage of distriscythe-bearing chariots about twenty. buting the measurements, which might be crowded upon this EXERCISE 71.-ENGLISH-GREEK. one line, upon other lines similarly placed for the same purpose. 1 Είς συνετος φιλος εστι κρειττων πολλων ασυνετων. 2. Ta en éßdoun Any further remarks will be made as we pzoceed with the method κοντα ποιει αμφι τας ημερας δισμυριας πεντε, πεντακόσιας και πεντηκοντα και of drawing the following problem :πεντε. 3. Ολος ο αριθμος της οδου απο της μαχης εν Βαβυλωνι εις τα Κοτύωρα PROBLEM XXXV. (Fig. 57).—Two slabs or rectangular blocks, jixo Zevopwvtos ou papetai otauoi ékat V CIKOPI, duo nupasangan cach of the same dimensions, 6 feet long, 4 feet broad, and I foot thick. εξακοσιοι εικοσι σταδια μυρια οκτακισχιλια εξακοσια, χρονου δε το πληθος | One block is above the eye, the other belou, esting on the ground; 4. Ο αριθμός του στρατευματος εστι τρισμυριοι ενι ακισχιλιοι | in every other respect the conditions of each are the same. Their 5. Exoc Tecoapes otparnyou tov otpa-ov ékatep's long sides are 40° with the PP; their nearest angles 3 fest to the των στρατιωτών τρισμυρίων εννακοσίων και € ηκοντα, 6. llapnear ev th left of the eye, and 2 feet within the PP. Height of the eye, 4 jax? Tpis TPECULUPtos kas é Fakto x.Acou éļakoo101 KAI ACVTIKOVTA TWY OTPUT.TW, feet, and distance of nearest angle to the eye 10 feet. The vertical και άρματα δρεπανηφορα αμφι εκατον και πεντηκοντα. space between the blocks is 6.feet. Our motive for employing two blocks of the same dimensions GEOMETRICAL PERSPECTIVE.—XII. and position, with the ove exception nained, is that we shall find it easier to explain ; and we hope our pupils, will mons clearly In proportion as the number of lines and angles increase, which understand the use of the PP when placed above the eye, and by compose the subject to be represented in perspective, so there which we intend to show that the proportions of the object can will follow a greater amount of working lines, drawn in various with equal capability be arranged upon a line above the ul, as directions from the picture plane. Under these circumstances upon one below it. By this use of two lines to represent the it will frequently be necessary to use more than one line to PP, the base of a column can be worked from the Pp below, and represent the pp, in order to prevent the confusion which must the capital from the one above. The same may ke observed occur when working all the details from one PP only. Therefore when representing windows, balconies, etc.,. in the upper storeys we are at liberty to use any number of lines as picture planes- of a large building. From is on the HL draw the semicircle an advantage fully DE E. DE(We appreciated by E have stated the every draughts distance of sight man when engaged in a way frequently in making highly done, in some of finished drawings Fig. 57 the military ernof very elaborate mination papers, subjects. The kind for the purpose of of work to which drawing attention these lessons are to it. It is said but an introduc that the distance tion, and which from the recarest must fall to the angle to the eye is lot of those who 10 feet, and that have studied per the object is 2 feet spective for some within or beyond practical purpose, the PP; therefore will not be reDVP31 the eye will be 8 stricted to cubes, feet from the PP, blocks of wood, which length will and the simple be the radius for objects we have deseribing the selected for our semieircle through practice, and to E.) The distance assist us in ex of the nearest an. plaining the prin PP gle of the object ciples. We know ] to the left of the the same rule for eye will be at b; c drawing a block in perspective is applied again in drawing a the nearest point of the object to the PP, from which lines church or a palace; but respecting the latter, that which must be drawn to both vanishing points; the perspective increases the labour, and not unfrequently perplexes the stu- lengths of c d and c e must be cut off by lines to their dent, is the increased amount and the great variety of details. respective distance points in the way already explained in We intend still to confine ourselves to simple examples, so Lesson IX., Vol. III., page 271. The line cd, which has been long as we have any new rule to give or fresh principle to drawn to vp', must be produced to the pp in h. The thickexplain ; let these be well learnt, then the application of ness of each block is 1 foot, that being added to the vertithem to more extensive and important subjects will be easy. cal space between them will be 8 feet; therefore the perpenWe now, therefore, introduce the practice of additional picture dicular line, or line of contact, must be 8 feet from 1 to i. planes, and that our explanations may, we trust, be clearer, Another Pp through i must be drawn parallel to the HL. Now, we will simplify the process by proposing a problem with as the blocks in this case are the same in their dimensions sad reference to two slabs or blocks only, of the same size, and each positions, the upper one could be very quickly and conveniently in the same position with regard to the PP. By this time our drawn from the lower one, by raising perpendicular lines from pupils will be prepared with the fact, that if an object touches the angles ; but we avoid this for a special reason that is, wo the picture plane its real length is represented upon the picture; wish our pupils to go through the construction again, upon and and as it retires from or beyond the picture, the space it occupies from the upper PP, in the same way as they did from the lower; f ul PP DE1 HL DE2 OVP2 6 the afterwards, a repetition of the process, when, in a future case, to the right of c d. Upon d m draw the plan of the lower the object above varies in size and form from the one below, the block; afterwards the plan of the upper one, e h i k; all its difficulties will not be so great. Probably it will be advisable to sides being one foot within the larger plan. In Fig. 59 we recapitulate some of the work, to prevent failure. Make a b have represented only the upper block; the lower one will be equal to the distance the object is to the left of the eye; draw simply a repetition of the one in Fig. 57, which our pupils 6 P8; make bf equal to the distance must not omit repeating when drawing the nearest angle is within the picture; PP Fig. 59. We will now commence with and because the line from b vanishes the HL, and proceed upwards. The at ps, therefore the line from f, to cut E, DE, PP, and Ps will be the same off the point within, must be drawn to as in Fig. 57. The distance of the DE', the distance point of E or Ps, to nearest angle a from the PP must be determine the nearest part of the ob Fig. 58, measured from b to c on the pp, and ject c. (Some writers on Perspective equal to psc, taken from Fig. 58. call the DE the DPS, meaning the dis. The distance c f, of the point a within, tance of the point of sight. It makes must be equal to f e (Fig. 58). Draw no practical difference, because Del or from a to vpl, and also the other way DEə represents the space between the to the prin m; a line from m perpeneye and the picture plane, that is, be dicularly to PP will be the line of contween E and PS; Ps being on the pic tact, upon which to measure the thickture plane, which is supposed to be in ness, m n, of the block. The length & perpendicular position; the line and breadth to be cut off on the lines below, marked PP, being its base.—See which vanish to vpl and vpo must be Fig. 21, Vol. II., page 360.) Through taken from the plan, viz., e h for c, directed from DVP', draw a line to the length, and e k for the breadth, T; make r s equal to the length of the as shown in e h and o k (Fig. 59). It block; draw from will be noticed & back again to that the differences DVP, which will of dimensions be-cut the vanishing Fig. 59. tween the two line from c to vpl blocks, and the in m; cm will greater distance then be the per. of the lower block spective represen from the PP, causes. tation of mk 16f 2 PP a change of posilength of the tion for the line of block. Through contact, or rather, C, directed from another line of DVP, draw a line contact must be into n; make no troduced. The perequal to the width pendicular from i of the block, and is the line of conrule from o back HL tact for the lower again to DVP2; vp2 DVP2 vel block, whilst the this will cut the one from m will be line from c to vp the line of contact in v; cv will be for the upper; the width of the Fig. 60, proving that in all block. We trust cases the first part A the remainder of of the construction the work, includ to be considered is ing the thickness the position of the of the block, will nearest point of present no difficul. the object, with ties. regard to the eye We will make and the PP; leavfurther use of this ing the rest to problem, by chang. whatever may reing the proportions sult from the work, of the upper block according to the to 4 feet long, and DE HL ovlea varied character PS DE 2 feet wide; its DO of the subject, and plan being in the the conditions centre of the plan given in the stateof the lower one. ment. In this case a plan Before we make must be drawn any further appliboth of the blocks cation of the rule and the PP, to show and process of the how the former are PP above problem, we situated and con will explain anonected with the latter, and from which we obtain the propor-ther important step connected with this part of our subject, tions and distances of the several parts from one another and afterwards combine the two in an especial case. and from the PP. Therefore Fig. 58 is the first considera Our next consideration will be the way in which we can make tion; it is a plan constructed according to the particulars use of a diagonal line for determining retiring distances and regiven in the question. Draw the PP. Anywhere, say from tiring proportions; that is, the angle which the diagonal makes a, draw the line a b, at an angle of 400 with the PP. Upon with the PP (we will suppose it to be the diagonal of a square). this last line find the point d, the nearest angle within the The diagonal is obtained by bisecting the angle formed by the P?; draw d c perpendicularly with the PP; place Ps 3 feet vanishing lines from a to vpl and vp> ; its vP and distance point DE PS n DP found, and in all respects treated as are the vanishing lines of 11. Find the length of the side of a square which is equal in retiring sides. area to the rectangle, the sides of which are 513 yards 1 foot PROBLEM XXXVI. (Fig. 60).—Two square slabs of different 11 inches and 1628 yards 11 inches. dimensions, the smaller of which is lying upon the other; the How much would it cost to cover the area with turf at 41d. plans of their centres coincide; the nearest angle of the lower per square yard ? one touches the PP. The side of the larger slab is 4:5 feet; the 12. If 100 lbs. of tea be bought at 45. 4d. and sold at 58., smaller, 3 feet. Thickness of each, 1 foot. Angle of sight, distance, and 100 lbs. of sugar bought at 6d. and sold at 78., what profit and height of the eye, as in the last problem. per cent. will be realised on the whole outlay ? A portion of the subject represented by the plan A must be 13. The removal of a quantity of brick earth, 32 square yards constructed, for the purpose of obtaining the length of that part in area and of a uniform depth of 2 yards, costs £2 2s. 80.; of the diagonal line between a and b. As the angles of the what is the cost of the removal of a cubic yard ? object are right angles, therefore the angle formed by the 14. A person's average annual expenditure, from the year vanishing lines from E to the HL will be a right angle. Bisect 1830 to the year 1850 inclusive, is £391 98. 2d. He finds that it by the line E 0; E O will then be the vanishing line of the in 1830 he spent £391 168., and in 1851, £445 8s. 9d. What diagonal of the slabs, and o the VP. Find its distance point by was his average annual expenditure from 1831 to 1851 indrawing from o the aro E DO. After the lower slab, mc de, clusive ? is drawn according to previous instructions, produce the per 15. In Austria 120 gulden (paper currency) are worth 100 pendicular m c through v; make mc and c v equal to the silver gulder. What amount of paper money should be obtained thickness of the slabs; in other words, mark their heights on the for £10 sterling if the value of £1 be 9 gulden 30 kreutzers in line of contact from m. Draw the diagonals m o, co, and silver (60 kreutzers 1 gulden)? V 0; also the diagonal d. Our object now is to determine the 16. What sum at £4 per cent. compound interest will amount nearest angle of the upper slab. Upon the diagonal of the base, in two years to £405 12s. ? m 0, we must cut off the distance of a b, in the plan A. Make 17. Å room is 60 feet long by 29 feet wide; how many people on n equal to the line a b, and from n draw a line to do, can be seated in it on chairs 11 feet wide, and placed 2 feet outting the retiring diagonal m o in h; mh will then be the apart from back to back, allowing a clear passage 3 feet wide perspective distance of a b. From h draw the perpendicular down the middle of the room and a space of 15 feet deep at one her; this perpendicular, cutting the diagonal from c, gives the end ? nearest angle of the upper slab in s; cv being the measured 18. Divide 296-293 by 41.967 so as to have six decimal places thickness of the upper slab, therefore s r is the perspective in the quotient. thickness. The diagonal d e, cutting the retiring base of the 19. Find the length of the longest chain in terms of which 88 upper slab from s each way, gives the perpendicular edges at 1 yards 2 feet 5 inches and 119 yards 2 feet 1 inch can both be and k. The remaining retiring lines must be directed to their expressed as integers. respective vanishing points. 20. By selling at 4} per cent. profit, a tradesman gained £47 14s. ; what was the prime cost of his goods ? 21. The interest on a certain sum of money for two years is LESSONS IN ARITHMETIC.XLVI. £71 16s. 7 d., and the discount on the same sum for the same EXERCISE 64.-MISCELLANEOUS EXAMPLES. time is £63 17s., simple interest being reckoned. Find the 1. Reduce to a proper fraction rate per cent. and the sum. 22. A barters some tea with B for flour which is worth 2} + 11 — 21 Х 2s. 3}d. a stone, but uses a false pound weight of 15€ oz. 3 + 2) — 41 過錯+ - 39.1 + $ What value should B set upon his flour that the exchange may 2. How many bricks 9 inches long, 4 broad, and 4 thick, be fair ? will be required for a wall 30 feet long, 20 feet high, and 4 feet 23. A garrison consisting of 2000 men, fixds during a siege thick, allowing 64 per cent. of the space for mortar ? that it has provisions for six weeks; at the end of a fortnight 3. When hay was £5 per ton, a well-to-do farmer hid him 400 men are killed in a sally. How long will the provisions last ? self in a load, and his weight, of course, was added in that of 24. Find the expense of painting a room 23 feet 6 inches the hay. Before the hay was shipped, the trick was detected; long, 12 feet 7 inches high, and 17 feet 6 inches broad, at and after another weighing, 7s. 6d. was deducted from the price. 3s. 6 d. a square yard. Find the farmer's weight. 25. The gold procured from Australia in nine months in 1851 4. A room is 20 feet long, 16 broad, and 12 high. If pure amounted to 313644 ounces. In 1861, the New Zealand gold. gold be worth £4 5s. per oz. Troy, and a cubic foot of gold fields yielded 314438 ounces in the same time. What is the weigh 19260 oz. avoirdupois, what is the value of the gold excess in weight and value (at £3 175. 104d. per ounce) of the which will exactly fill the room ? average monthly return from New Zealand over that from 5. A wine-merchant buys 12 dozen of port at 84s. per dozen, Australia ? and 60 dozen more at 48s. per dozen; he mixes them, and sells 26. Find the difference between the amount of £247 10s. for the mixture at 72s. per dozen : what profit per cent. does he two years, and the present worth of the same sum due after two realise on his original outlay? years at 5 per cent. 6. How many cubes of which the sides are 27 inches can be 27. How many bricks, of which the length, breadth, and cut out of a cube of which the side is 22 inches ? thickness are 9, 6,3 inches respectively, will be required to build 7. A young lady desires to paper her room with postage- a wall whereof the length, height, and thickness are 72, 8, and stamps, but being herself unable to calculate the number which 14 feet? will be required, she supplies the following data :-Her room is 28. Express as the fraction of £10 the difference between 14 feet 9 inches long, 9 feet 3 inches broad, and 10 feet 6 £8] and £8 x }; and find the value of į of a ton of sugar inches high ; it contains two windows, each 5, feet by 4 feet, when of a ton is worth £6 5s. and three doors, each 6 feet by 3 feet; a postage-stamp is li of 29. Find two numbers whose greatest common measure is an inch long, and of an inch broad. Make the calculation for 179, and least common multiple 56385. her. 30. In which way had one better buy sugar, at 3 guineas per 8. The daily issue of the Times is 60,000 copies. Three days cwt., or at £2 16s. 4d. per quintal of 100 lbs. ? and how much of the week it consists of 3 sheets, and for the remaining three is one buying when the gain by the more advantageous way is a of 4 sheets. If a sheet be 3 feet long and 2 feet broad, find the guinea ? number of acres which the weekly issue of the Times would 31. On what sum is the daily interest at 4 per cent. one cover. penny ? 9. Express of 78. 6d. + 625 of 10s. --545 of 9s. 2d. as a 32. If 16 darics make 17 guineas, 19 guineas make 24 pistoles, decimal fraction of £10. 31 pistoles make 38 sequins, how many sequins are there in 1851 10. Among how many men must £73 be divided in order that darics ? the share of each may be £4 178. 4d. ? And among how many 33. Which way had one better buy coffee, at 6 guineas a cwt., must £79 178. 6d.' be divided in order that half of them may or at £5 123. 4d. per quintal of 100 lbs. ? And how much is ons have 10s. 7d. each, and the other half 7s. 2d. each ? bnging, when the loss on the less advantageous way is £1? |