of sequence of sounds in the English language as we find it in books. Of these letters k, m, p, b are commonly initial letters in words and syllables, and do not require to be abbreviated. 79. The above calculation shows what letters ought to have abbreviated forms. They are the first half of the list, omitting k, m, p, b, for the reason given. In the second half of the list, the letters are so unfrequent that it is not necessary to accommodate them with any other signs than their alphabetic forms. The observance of this rule will show when a vowel, if not inserted, is to be read before -tion, and when before the previous consonant. After simple tor d, the -tion hook is always written on the right; thus, addition. 92. The n,f, and -tion hooks may be used in the middle of a word; as, render, \ perfect, national; and the -tion hook derision, vision. 80. S and are provided for by a circle (see paragraph 17). I may be thickened for -sion; thus, and are abbreviated by means of the hooked letters explained in paragraph 47. In the remaining part of this lesson we shall show how to abbreviate n, f, and v, the termination tion, t and d, the combinations st and str, both initial and final, and the syllables ther, ter, der. We shall then have given, in addition to the shorthand alphabetic form of each letter, a still briefer form for the most frequent half of the letters. The remaining letters of the alphabetnamely, sh, ith, j, the, ch, g, w, h, ing, zh, y-all put together, do not occur more than half as frequently as any one of the letters s, n, r, t, and therefore do not require abbreviation. N HOOK. 81. N following a STRAIGHT letter is expressed by a small final book on the LEFT-HAND side of a perpendicular stroke, and on the corresponding side of any other straight letter (see Table of Consonants in Lesson V.), thus 82. By making the hook into a circle, s or z is added; thus J- tones, J-trains, pains, Tocanes, turns, explains When the s or z circle This circle on the left means ns only when it expences. 83. The ns circle is made double-size for nsez; thus, feign, Clean, ❤ known, mine. 85. S or z is added to the n hook following a curve, by making a small circle at the end of the hook; thus, mines. HALVING PRINCIPLE, ADDING T OR D. 93. By halving any consonant (see par. 97), whether single, double, or treble, t or d is added, according as the letter is thin or thick; t being generally added when the letter is thin, and d when it is thick (see last column of Table of Consonants in Lesson V.); thus, talk, talked, bake, baked, rip,^ ripped; live, lived, ▲ rub, ^ rubbed, \___ beg, \__ begged. 94. A vowel BEFORE a half-sized consenant is read before both letters; as,.) east, .) eased, \ oft, ached, - act. 95. A vowel AFTER a half-sized consonant is read next to the primary single, double, or treble letter; thus coughed, point, Crossthwaite, bread, 9. street. 96. M, n, 1, and r are shortened for the addition of t, and these J- tone, J·train, \ pain, → cane, turn, explain. shortened strokes, when thickened, represent md, nd, ld, rd; thus↓ tempt, L. deemed, sent, send, old, appeared. the addition of t or d are 97. The only consonants that do not admit of being halved for darew, ng, mp, and Ir; but they may be halved when ending with a hook, as, impend. 98. Lt, when standing alone, is written upward; in other cases, either upward or downward: Id is always written downward; thuslute, melt, \ pelt, knelt, fold. 99. Dis added to both light and heavy letters, for the past tense; as, 1. melted, peopled, ordered, measured. 100. To express d instead of t after the n hook (also to distinguish v from f), the hook is thickened, but this is not important; thus♫• attained, I pained, ß? lend, ¶, strife, q↓ strive. 101. The practised phonographer may use a half-sized letter to represent either an added t or d; as, mind, ✈ upward, downward, forward (wd being used as a contraction for ward), rapid, afford, alphabet, between. 102. A full-sized and a half-sized consonant, or two half-sized consonants, should not be joined unless they form an angle at the point of union; because it would sometimes be doubtful whether such combinations were meant for a single letter, or a full-sized and a half-sized letter, or two full-sized letters. For instance, k and kt, (upward) and kt, tr and tt, d and tt, t and nt, nt and mt, are not allowable combinations: these double consonants should either be 86. A vowel after a letter with the n hook is to be read BEFORE the n; thus, men. To express a vowel AFTER n, the stroke n must be used; thus, many,tiny. As the stroke n (and also the strokes, and frequently t and d), at the end of a word, thus indicates a final vowel, the vowel may be omitted in swift writing. FOR V HOOK. 87. For v, when following a STRAIGHT letter, is expressed by a final hook on the RIGHT-HAND side of a perpendicular stroke, and on the corresponding side of any other straight letter; thus— | tf, \ chf, \ pf, kf,rf (upward), as in tough, David, cough, There is no f or v hook to CURVED letters. -TION HOOK. lithograph, hf; resolved into their simple letters, or the pen should be taken off; thus 88. The termination -tion (shon) is expressed by a LARGE final art, elt, hook; thus, fashion, mission, version, nations. 89. When the -tion hook follows a CURVE, it is written on the INNER side, like the final n hook. It may be written on EITHER side of a STRAIGHT LINE, under certain restrictions, as follows: 90. At the end of a straight letter beginning with a hook or circle, or springing from a curve, the -tion hook, when final, is written on the OPPOSITE side, that the straightness of the letter may be preserved; thus, oppression, collection, secretion, practicable. length consonants are named -ket, gent, emt, emd, wilt, etc. ST AND STR LOOPS. 104. St is written by a loop about half as long as a consonant; as, •f state, I stop, stock, ♂ steam, 6 still, - toast, kissed, fast, last, rest. 105. This loop is used chiefly as initial or final, but it may be employed medially when the loop can be distinctly formed; as in du av testify, justify, investigation. 106. A loop about two-thirds as long as a consonant represents str; thus, faster, ≈ muster, minister. The practice of writing the str loop INITIALLY is not recommended:strap cannot be halved for the past tense: write strap, ¶ strapped. १ and the 107. These loops may be combined with the initial pr final pn series of straight letters; thus— ¦ stoker, → against, ŏ punster. ward r), reconcilable (upward r, the two forms of r are used, 108. S may be added to a final loop or to the large ss circle by in accordance with a rule which will be given hereafter), T miscon continuing the stroke of the loop or circle; thus 109. A hook made by continuing the s circle to the other side of the consonant, adds -tion (shn) or -sion (zh); thus and consented, and compare, and connected, has commenced, the sign for and, in the next Lesson.) and is content. (See 110. The circle s or z may be added to this back hook, and it inherit, may occasionally be used in the middle of a word; thus mother, stronger, letter, distemper. inhuman. This principle of abbreviation cannot safely be used before l in insolvent, insoluble, etc., which, if written insolvent, would not be sufficiently distinct from solvent. 125. A prefix resembling in sound any of the above may be written in the same manner; thus, accompany (akompani), neither, inventor, father, order, 112. These double-length consonants may be hooked for n, as slander. They are vocalised like the half-length ones (par. 94, 95). They should not be adopted by the learner until he has gained some degree of fluency in writing. VOCALISATION OF THE PL, PR SERIES OF CONSONANTS. 113. One of the long vowels ah, eh, ee, may be expressed BETWEEN the two letters of one of the pl, pr series of consonants, by a small circle BEFORE OF ABOVE the consonant; thus, I dear, careless. 114. The SHORT vowels a, e, i, are written by a small circle placed AFTER OF UNDER the cousonant; as 129. Ly as heavenly. This affix does not interfere with ality. It is generally more convenient to join the ; as goodly. A separate ly may be omitted in reporting. tality; 130. MENTAL, MENTALITY as instrumental or instrumen- O SELVES, as themselves. 132. SHIP as stewardship, horsemanship. Sometimes the two letters sh, p, can be written faster, if joined, than a separate sh; thus, friendship. 133. A logogram (or word-letter) may be used either as a prefix or affix; thus, Lordship, afternoon, undertake, advantageous, hereafter. 120. CON or COM is expressed by a light dot, written at the beginLESSONS IN LATIN.—XXXIV. ning of the word; thus, J. contain, v comply. When preceded by DEVIATIONS FROM THE MODEL CONJUGATIONS. a consonant, either in the same or the preceding word, con or com THE four conjugations are only so many classes into which is understood by writing the syllable that follows, UNDER or CLOSE Latin verbs are put. These classes are determined by the tokens already stated, and the verbs in them are formed acTo the consonant that precedes; thus, inconstant, accom- cording to the models already given. But the verbs which plish (in practice, the vowel of ac may be omitted), decompose, make up these classes do not exhaust the whole stock of Latin verbs. There are others that more or less depart from the discompose, by discontent, irreconcilable (down- models. The verbs that depart from the models may in general be ranked under three descriptions. 1. They differ from the model verbs in the perfect and the supine. 2. They differ from the model verbs in other parts. 3. They lack several parts which are possessed by the model verbs. As these three classes do not conform to the rule or standard afforded in the four conjugations, they might all be termed irregular. The epithet irregular, however, is commonly applied only to the second class, and as no good reason compels us to depart from the ordinary usage in this case, we shall specially apply the denomination of "the irregular verbs" to those verbs which differ from the models in other parts than the perfect and the supine. The verbs which differ in the perfect and the supine may be called deviational, as in a marked way deviating (de, from, and via, a way) from the ordinary forms; and those that are without certain parts may be termed defective (de, from, and facio, I make). Thus we obtain four classes of verbs:1. The regular, or those which mainly follow the paradigms of the four conjugations. 2. The deviational, those that depart from those paradigms in the perfect tense and the supine. 3. The irregular, those that depart from the paradigms in other parts. 4. The defective, those that are wanting in mood, tense, and person. The verbs that follow the forms of the four conjugations are dignified with the term regular, because such verbs exist in greater number than any other class. The same facts may be set forth in this VIEW OF THE LATIN VERBS. Regular. The Four Conjugations. Irregular. 1. Deviational. 2. Irregular. 3. Defective. The departure from the models would be very much lessened, were we at liberty to enter at length into the processes by which the present tenses now in use were formed, and exhibit the real roots or stems of the several verbs. In the restricted space, however, here at our disposal we can only give one or two instances. Take sino, I suffer, or allow. You think the stem to be sin. It is not so; the n is introduced into the present merely for the sake of strength in pronunciation. The stem is si. Sio would be a weak sound, two vowels coming together; and the Latins preferred saying sino. From si, however, is regularly formed both the perfect and the supine, thus: si, si-vi, si-tum. Take rumpo. Here, to obtain the stem, you must elide the m (that is, n which before p becomes m), and you have rup, from which are regularly formed rup-i and rup-tum. In cresco, you must in the same way leave out sc, and thus getting as the stem cre, you form the perfect by adding vi; for example-cre-vi. I subjoin similar instances: sue-sc-o, sue-vi; pa-sc-o, pa-vi; no-sc-o, no-vi; so the supines, cre-tum, sue-tum, pa-stum (the s of the present is retained before the t), no-tum. Disco represents a peculiar class of verbs; namely, those that form the perfect by what is called reduplication; that is, the repetition of the initial consonant with a connecting vowel, thus: disco, discere, didici; the root is properly die; di(c)-sc-o, di-dic-i. Having given these intimations, and inviting our pupils in due time to follow the thread, we now resume the order of the four conjugations, and shall set forth the principal deviations from the model forms. I. DEVIATIONS IN THE FIRST CONJUGATION. 1. The Perfect with Reduplication. i. Do, dědi, dăre, dătum, to give; the a is short in the stems, as dăbam, dăbo, dărem, except das and då. According to do form the compounds of do, of which the first part is a dissyllable, as circumdo, circumdědi, circumdatum, circumdure. The compounds of do having monosyllables as prefixes follow the third conjugation, as addo, addere, addidi, additum. It is not uncommon for a verb of one conjugation thus to pass into another conjugation. ii. Sto, stěti, stare, statum, to stand (with abl. to cost). The compounds with monosyllables bave in the perfect stiti, as consto (I consist of), constiti, constare; participle future active, præstaturus, constaturus, obstaturus, etc. The compounds with dissyllables retain stěti, and have neither supine nor par ticiple future active; also circumsto (I stand around), circumstěti. 2. Multo 1. Deus nobis dedit animum, quo nihil est præstantius. sanguine nobis victoria stetit (cost, so we say stood in). 3. Mater omnium bonarum artium sapientia est, quâ nihil a Deo immortali uberius, nihil præstabilius hominum vitæ datum est. 4. Deus corpus, 5. ut quandam vestem, animo circumdedit, et vestivit extrinsecus. Quorum patres aut majores aliquâ gloriâ præstiterunt, (ii) student 6. Parentes carissimos plerumque eodem in genere laudis excellere. habere debemus, quod ab iis nobis vita tradita est. 7. Non dedit beneficium, qui invitus profuit. 8. Quinam magis sunt tui quam (ii) quibus tu salutem insperantibus reddidisti? 9. Cives acerrimos propugnatores libertatis se præstiterunt. 10. Ingens hominum multitudo oratorem in foro circumstetit. 11. Eloquentia ad salutem hominum data est a naturâ. 12. Eloquentia ad conservationem hominum data est a naturà. 13. Malus orator eloquentiam ad bonorum pestem, per14. Quid est tam inhumanum quam eloquenniciemque convertit. tiam, a naturà ad salutem hominum et ad conservationem datam, ad bonorum pestem, perniciemque convertère? 15. Stipendium ex longo tempore militibus non erat datum. 16. Seditio inter milites orta est. 17. Quum stipendium ex longo tempore non esset datum, seditio inter milites orta est. 18. Tu mi amice, mihi fidem præstaturus es. 19. Certo scio te mi amice, mihi fidem præstaturum esse. 20. Nihil nobis obstat. 21. Nihil nobis obstaturum est. 22. Victoriam adipiscamur. 23. Credo, nihil nobis obstaturum esse, quominus victoriam adipiscamur. 25. Non 24. Victoria constitit multorum fortium virorum morte. dubitabamus quin multorum virorum fortium morte victoria constatura esset. 26. Perstaturus ne es in sententiâ tuá ? 27. Nescio perstaturus ne sis in sententià tuâ. EXERCISE 126.-ENGLISH-LATIN. 1. He gave the general a fleet. 2. He will give thee a fleet. 3. Dost thou think that he is about to give a fleet to my brother? 4. Nothing has cost men more (pluris) than avarice. 5. God has given me a sister, than whom nothing is dearer to me. 6. My sister will show herself loving to me all her life (acc, with per). 7. The soldiers showed themselves very brave, but the victory cost the death of many brave men. 8. Nothing hinders our guining the victory (hinders, lest we should gain, stands in the way to prevent our gaining the victory). 9. I believe we shall gain the victory. 10. Socrates surpassed all philosophers. 11. Who does not know that Socrates surpassed all philosophers? 12. Dost thou believe that thy son will surpass all his equals? 13. A great multitude surrounds the orator. 14. Pay has not been given to the soldiers. 15. I will give pay to the soldiers. 16. Take care lest sedition arise among the soldiers. 17. Wilt thou persist in thy opinion? 18. I do not know whether I shall persist in my opinion. It may be desirable to illustrate more fully the construction of circumdare and quominus. CONSTRUCTION OF "CIRCUMDO." 1. Aliquid alicui rei, that is, a nominative or accusative of the thing with a dative of the thing. 2. Aliquem or aliquid aliquâ re, that is, an accusative of the person or thing, with an ablative of the thing. 1. "Aer omnibus est rebus circumdatus."-Lucretius. Iit. Trans. :-" The air to all is to things surrounded." Id. Trans." All things are surrounded with air." 1. "Tectis ignes circumdatos restinximus."-Cicero. Lit. Trans. :-" To the roof fires put round we extinguished.” Id. Trans. :-" We extinguished the fires which had been set to the Fenses." 2. " Oppidum vello et fossà circumdedi."--Cicero. Lit. Trans. :-"The town with a rampart and ditch I have surrounded." Id. Trans. :-" I have surrounded the town with a rampart and ditch." "Circumdato me brachiis."-Plautus. 2. Lit. Trans. :-" Surround me with arms." CONSTRUCTION OF "QUOMINUS." Quominus is made up of quo, in order that or so that; and minus, less (least, lest) or not; consequently, quominus is equivalent to so that not; a more simple way is to render it by to prevent. Quominus takes a subjunctive mood. If you use "to prevent" you must in English use also the idiomatic construction of "to prevent;" but in putting that construction back into Latin and so employing quominus, you must employ, of course, the proper Latin construction; as in the following example: "Nihil de me tulistis quominus in civium essem numero."-Cicero. Lit. Trans. "Nothing from me you bore so that not in of the citizens number I should be." Id. Trans.:-" You got nothing from me to prevent my being in the number of the citizens." EXERCISE 127. LATIN-ENGLISH. 1. Nihil impedit quominus id quod maxime placeat facere possimus. 2. Non repugnabo quominus omnia legat. 3. Mors non deterret sapientem quominus reipublicæ consulat. 4. Interpellent me quominus Lonoratus sim, dum ne interpellent quominus respublica a me commode administrari possit. 5. Nemini civi ulli, quominus adesset, satis justa excusatio est visa (no pretext appeared sufficient to excuse any citizen from being present). 6. Fossam Latam lecto circumdedit. Exercitum circumdat hostium castris. 8. Circumdat sibi milites. Circumdabit brachia collo tuo. 10. Circumdedit urbem tumulo. Extra hos cancellos egredi conabor quos mihi ipse circumdedi. Egregiam famam paci circumdedit (gave to, invested with). 13. Fautores illi hanc famam circumdederunt. EXERCISE 128.-ENGLISH-LATIN. 7. 9. 11. 12. 7. While we attempted to assault the city, the enemies attacked us from behind. 8. Each one measures dangers by his own fear. 9. The wise man both remembers past things with pleasure, and so possesses present things as to perceive how great and how pleasant they are. 10. Take care you do not obtain honours by flattery. 11. Orators, before they begin, premise certain things. 12. In all matters, before we begin, we should employ careful preparation. 13. All the citizens had adorned and clothed their houses with flowers and wreaths, because they were waiting for the king. 14. While the hostile army was demolishing the private and public buildings of the city, the citizens were filled with the greatest grief 15. While the enemies were dividing the spoil amongst themselves, we charged them with the greatest impetuosity. 16. The general exhorted the soldiers, that they should try all methods by which they might relieve the city from. the blockade. EXERCISE 120.-ENGLISH-LATIN. 1. Sæva tempestas coorta est. 2. Sæva tempestas coorietur. 3. Sæva tempestas cooritur. 4. Sævæ tempestates cooriebantur. 5. Sæva tempestas cooriebatur. 6. Nautæ multos labores experti sunt. 7. Hostes domum tuam demolientur. 8. Meam sororem opperiar. 9. Mater mea heri me opperiebatur. 10. Illi eblanditi sunt honores. 11. 12. Nolo honores eblandiri. 13. Prius quam Honores eblandierisne ? ordiaris, adhibenda tibi est industria. 14. Tellure potitur. 15. Omni urbe potitus est. 16. Libris patris mei potiar. 17. Mi puer, cave ne mentiaris. 18. Solum mali mentiuntur. 19. Malum est mentiri. 20. O pater, nunquam mentiar. 21. Illi mentiti et puniti sunt. 22. Turpe est mentiri. 23. Sol et bonis et malis oritur, tanta bonitas Deo est 21. Rex fortibus militibus honores largitur. 25. Inter duo filios bona partitus est. 26. Quicquid a tellure oritur, venit a manu divinâ, EXERCISE 121.-LATIN-ENGLISH. 1. Those are to be laughed at who teach others that which they themselves have not tried. 2. Every animal loves itself, and as soon 3. Nature has bestowed so as it appears, aims at preserving itself. great a productive power on the conveniences and necessaries of men, in order that those things which are begotten may seem to have been given to us advisedly, and not to have arisen by chance. 4. Herodotus passed through many lands, and certainly related many marvellous things, but these he himself invented not, but others (invented them) from whom he heard them. 5. Now for three months we had been waiting for a friend, when his death was announced to us. 6. Sulla suddenly appeared to the Romans, and began a most frightful civil 7. A wise man will never flatter bad men, never invent any false thing, never occasion injury to others. 8. If we attack the enemy quickly, there is no doubt that in a short time we shall get possession of the city. 9. As soon as the sun has risen, we shall set out. 10. Take care that you do not flatter bad men. 11. The enemies hastened to get possession of the city. 12. An even number is easy to divide. 13. The sun rising, we set out. 14. A fierce tempest having arisen, great fear seized all the sailors. 15. With the greatest pleasure we behold the sun when about to rise. war. 1. Nothing hinders you from being a good boy. 2. I will surround thee with fame. 3. He will put a garment round me. surrounded his sister with honour. 5. Surround the city with fire to sunt. 6. That is no excuse for your prevent the citizens from going out, absence. EXERCISE 122.-ENGLISH-LATIN. 1. Sole oriente, tenebræ diffugiunt. 4. He has 2. Solem oriturum magná cum voluptate specto. 3. Coortâ tempestate, naves nostræ sparsæ 4. Nulli calamitatem molior, non etiam malis. 5. Ortusne est sol ? 6. Horá octavâ sol orietur. 7. Repente hostem adortus est. 8. Meam ordiar orationem. 9. Orator orationem ordiebatur, quum judex intrabat. 10. Non est dubium quin tuis potituri sitis. 11. Simulatque nati sumus, movemus. 12. Omnes homines se ipsos diligunt, ac simulatque facultatibus potiti sunt, inter se eas partiri debent. 13. Inter egentes facultates partitus est. 14. Multi Britanniam emensi ignorant quam sit felix potensque. 15. Spero te nunquam mentiturum esse. 16. Tempestas orietur. 17. Omnes existimant, tempestatem KEY TO EXERCISES IN LESSONS IN LATIN.-XXXIII. EXERCISE 117.-LATIN-ENGLISH. 1. Cæsar feared to oppose his army to a river of such magnitude. 2. Cæsar is of opinion that it should be attempted, and a trial made. 3. Cæsar, although he feared to oppose his army to a stream of such magnitude, yet is of opinion that it should be attempted and a trial made. 4. Nor did they fear that they should be surrounded. 5. They said that they feared not the enemy. 6. They said that they feared the narrow road. 7. They said that they feared that corn was not supplied conveniently enough. 8. They said that they feared not the enemy, but the narrow road; and that they were afraid that provisions could not be supplied conveniently enough. 9. Be not afraid, the ship is safe. 10. I fear greatly on account of the republic. EXERCISE 118.-ENGLISH-LATIN. 1. Timeo ne operam perdideris. 2. Timeo ne domus cadat. 3. Timeo ut frumentum in urbem supportetur. 4. Dux metuebat ut exercitus ejus veniret. 5. De suâ puellulà pulchrà metuunt. 6. De fortuna tui nihil timeo. 7. Rex ducesque timent ne circumveniantur. 8. Cicero experiendum judicat. 9. Timeo ut experiri possit. EXERCISE 119.-LATIN-ENGLISH. coorituram esse. men. EXERCISE 123.-LATIN-ENGLISH. 1. The arts support themselves. 2. We shall always pity wretched 3. When you are sick, you ought to comply with the instructions of the physician. 4. Foolish men perceive the faults of others, (but) forget their own. 5. The principal charm of youth springs from modesty. 6. O boys, reverence old age. 7. O boy, I confess the truth. 8. Take pity on the destitute. 9. Let the scholars reverence their teachers. 10. I doubt not that you will promise me your protection. 11. We behold with great pleasure the noble examples of virtue which have been recorded in history. 12. Who knows not how many abuse eloquence ? 13. We have enjoyed peace during many 14. All the citizens fear that the enemies will attack the city. 15. As soon as we have arisen, we proceed to our business. 16. The citizens, having obtained liberty, will enjoy the greatest pleasure. 17. Aid the fallen. 18. Converse as boldly with your friend as with yourself. 19. Be not angry with those whom you ought to love. 20. If we always pursue the path of virtue, access to heaven will sometimes lie open to us. 21. Perform your part well. 22. By concord, small things increase; by discord, the greatest things fall to pieces. 23 Glory, like a shadow, follows virtue. years. (For Key to Exercise 124, see next Lesson.) perspective of the slab. We will now draw the same slab not having any of its sides parallel with the PP. PROBLEM XXXIV. (Fig. 56).-The slab of the last problem, having the same dimensions; the position only different; one of its edges is at an angle of 40° with the PP; the remaining conditions as before. GEOMETRICAL PERSPECTIVE.-XI. PROBLEM XXXIII. (Fig. 55).-A stone slab is inclined at an angle of 35° with the ground; it is resting on an edge of one of its ends, which is parallel with the picture plane; the edge on the ground is 4 feet within; length of slab 9 feet; breadth 8 feet; thickness 2.5 feet. Height of the eye 4 feet; and distance from the PP 8 feet. Although the slab is on an incline, yet its ends are parallel with the PP, there fore it may be considered a parallel case of perspec tive; had it been flat on the ground, its vanishing point would have been the PS; but being an incline of 35° its VP is perpendicularly above the PS, found by drawing a line at 35° with the HL from DE (we hope our pupils now clearly understand that the vanishing lines for inclinations are always drawn from the DP of the VP, to which they would retire if they were horizontal; we beg them to turn back and examine the figures of the previous problems to confirm this). The first thing to be done is to find the point b within the picture; make ca and cd each equal to 4 feet (these distances are together equal to the width of the slab); and as b is 4 feet within we will use the point a again, and rule a line to DE; where this line cuts the line c PS gives the point b: join d PS and a PS; through b draw e f. Now, as the face and end of the stone are at right angles with each other, it is very evident that if the vanishing line of the face is at vr1, the vanishing line of the end must form a right angle with it and terminate at VP2. Find VP2, and its distance point DVP2; through d draw the perpendicular hd gmi. Now we must draw the end of the stone o e pf thus: because it vanishes at VP2, therefore we must draw a line from DVP2 through e to the measuring line in h; mark off the thickness of the stone 2.5 feet from h to g, and rule g back again to DVP2, this gives the perspective thickness of the slab o e directed to VP2. To obtain the opposite corners p, ƒ will present no difficulty; from e draw to VPI, and from the distance point of vel draw a line through e tom on the measuring line; make mi equal to the length of the slab; a line from i to DVP will cut the vanishing line e to vpl in n; draw n s towards VP2, and st meeting a line from p to the VP', this will complete the Draw the HL, BP, distance E and semicircle. Find the VP for the end, viz. VP, by a line from E at an angle of 40° with the tangent line. VP2 is found by drawing a line at a right angle with E VP'; draw E PS b; bc is the distance of the nearest point within, determining a (remember DE is the distance point for cutting the line b Ps to find a). The vanishing point for the face of the slab will be VP3, found by drawing a line from DVP at an angle of 30°; VF will be the vanishing point for the end. For their distance points, draw from VP', with a radius to DVP2, an arc to DVP. From VP3, with a radius to DVP2, draw an are to DVP3. Το draw the horizontal edge a o, draw a line from a to vp1; draw from DVP through a to g; make g h equal to the width; from h rule back again to DVP, giving the required length of the end a o. Through a draw from DVP to e on the measuring line, ef is equal to the thickness of the slab; draw from f to DVP', and through a directed to VP draw a n; draw from n to vp. Through o draw o p directed to VP; this will be the end of the slab. Draw from a to VP; through a from DVP3 draw a line to i; makeim equal to the length of the slab; draw from m back again to DVP3, this will produce a r. Draw a line from n to VP3 for the inclined edge. A line directed from VP through r, meeting the line from n to vp3 in s, will be the termination of the length. From s to t, directed to vp1, will be the upper edge of the face of the slab. If these two problems upon the same slab, in the same position, and having the same dimensions, but viewed from different points, are well studied, with regard to that especial reason which suggested their introduction-viz., the principle of finding vanishing points for inclined lines and planes, and the method of treating them according to the characters and proportions of the objects, and the view w have of them-they will help to make our future problems po sessing more details easy to be understood. |