GEOMETRICAL PERSPECTIVE.-XXI. SHADOWS CAUSED BY ARTIFICIAL LIGHTS-CONCLUSION. WE now intend to prescribe rules for projecting shadows ef is the shadow of the post e m. The same process is to be the wall A, draw a perpendicular line to cut the trace of the plane of shade in n; draw from n through b to g on the intersection of the two walls; through g, directed from e, draw g k; will be found to meet a line drawn from ƒ through c; ckgb will be the shadow of the pole. PROBLEM LXII. (Fig. 101).-A ladder leaning against a wall casts its shadow partly on the wall c; it is continued upon vary, as well as the surfaces upon which the shadows are Fig. 99. plane of shade must be drawn by ruling a line through the luminary, the vanishing point of the plane receiving the shadow, and the trace of the plane which casts the shadows. PROBLEM LX. (Fig. 99).-A street lamp surrounded by an iron fence. Draw the perspective projection of a circle according to the diameter a b; place the posts at pleasure (in the figure they are situated upon the lines by which the circle is produced). The post c d represents the given height of the whole, through the top of which is drawn another perspective circle, in order to obtain the proportionate heights of the remaining posts, and the upper edge of the fence. From the foot of the lamp-post lines are drawn outwardly to meet the rays drawn from the light in the lamp at o, and through the top of each post as of; an open door, not perpendicular with its connecting wall; and lastly upon the ground. Project also the shadow of the door. Let A be the source of light. After the last problem relating to the pole, it will not be difficult to understand the shadow from the ladder in this case; the position of the door will cause some difference in the course of the shadow which falls upon it. The method of projecting the shadow from the top of the ladder at a on the opposite b wall c, is the same as that of the pole; for this portion vp is the vanishing point on the trace of the plane of shade; VP2 is the vanishing point of the doorthat is, for the lines b c and d e. Το find the vanishing point of the shadow on the door, draw a perpendicular line to meet the trace of the plane of shade from o, where the vanishing line from the bottom of the door intersects PS B; this will produce vp; lines drawn upon the door in continuation of those from a, and directed by VP, will be those of the shadow required. And lastly, lines drawn from B through the foot of the ladder will unite with those on the door at the base. To project the shadows of the steps, draw lines from A, the luminary, through the extremities of each step to meet the shadow lines of the sides of the ladder between which the shadows will be projected. To produce the shadow of the door on the wall D to the left, let h n be the angle of the wall, and hg its base. Draw f e, produced to k; from k also an indefinite perpendicular line through s; the rosition of s will be explained presently; produce d e, the base of the door, directed by its vanishing point vp2, to intersect the wall D at g; draw cb to p, also the perpendicular line g p, then the plane of the door will intersect the plane of the wall in g p; join f, the base of the luminary, with the vanishing point of the door vp2; this will intersect the base of the wall at m; draw a perpendicular line from m to VP5; this will be the vanishing point for the shadow of the door on the wall c. Directed by VP5, and through b, draw br; also, directed by p, draw s r. The portion of the shadow r s k h will be on the wall D, and the remainder on c. In these lessons on Perspective we have endeavoured to explain principles, rather than multiply examples; and in order to carry out our intention, the subjects we have chosen have been those that would enable us to employ rules having a In conclusion, for whom, it may be asked, are these lessons in Perspective intended? Are they to be studied only by architects and mechanical draughtsmen ? Undoubtedly these men have the first interest in them, as they satisfactorily provide a means for rendering their work more truthful and intelligible. An architect and a painter may, in many respects, be famous in their respective professions; but if they are ignorant of the first principles of design, which are founded upon the indisputable rules of Perspective, it is hardly possible they can have that full power and freedom of expressing their ideas, which is so essentially necessary for the success of their work. To the architect a perspective drawing of a building from some particular point of view, showing how it will appear when erected, will answer in every respect the purpose of a model; while an intimate acquaintance with angular perspective will relieve the painter general application. That which is of the first importance in Perspective, and may be considered the foundation upon which the whole science is built, is the projection of a point, or a series of points, which, when united by lines, represent, according to the positions of the points, the object required; consequently there can be but few. especial rules. The infinite variety of objects the draughtsman has to represent, with their numberless forms and positions, may sometimes perplex him; but however complicated they may be, experience will teach him that when in difficulties he must invariably fall back upon first principles. We have thought it advisable, in stating our problems, to give them under relative proportions, and to employ a scale of measurement. Practically this will be found to be of great advantage, as otherwise we should have given but abstract forms, which might assist in explaining a theory, but for any useful purpose would in a great measure frustrate the end we have always kept in view-namely, that of making these lessons really serviceable to our pupils. from the old, worn-out, and only resource of those who understand but one vanishing point, the point of sight. But there are others who have no professional necessity for studying Perspective, to whom a knowledge of its principles is as important as it is to those who are called upon to practise it; believing it to be, as we most certainly do, one of the most necessary and important chapters in the grammar of art. No one will venture to maintain that a thorough command of a language can be acquired without a very close study of its construction, knowing full well what the results would be; and for the same reason it would be dangerous for any one to pass judgment upon works of art, if ignorant of the principles of that art. How many there are who do this, and give their opinions in the most self-satisfied way on points of art of all sorts! But if we mention only one inducement out of many we might propose, for studying Perspective as it ought to be studied, practically, it would be that it enables us to understand some of the beauties of art, to know its capabilities, and to enter into its enjoyments. LESSONS IN ITALIAN.-XXIV. CONJUGATION OF THE AUXILIARY VERB A-VÉ-RE, to have. WE now proceed to bring under the learner's notice the auxiliary verb avére, to have, adding remarks on the various peculiarities in each mood, as in the case of the auxiliary verb essere, to be. I. INDEFINITE MOOD. Present.-A-vé-re, to have. Preterite.-A-vé-re a-vú-to, to have had. Future.-A-vê-re ad a-vé-re, ês-se-re per a-vé-re, to be about to have. Preterite Gerund.-A-vên-do a-vú-to, having had. [about to have. Conditional Present. Sing. A-vrê-i or a-vrí-a, I should A-vré-sti. [have. A-vrêb-be or a-vrí-a. Plur. A-vrém-mo,10 A-vré-ste. [or a-vrí-a-no. Sing. A-vrê-i a-vú-to, I should have Plur. A-vrém-mon-vú to. I. REMARKS ON THE INDEFINITE MOOD. [had. | I have, are regular, but now obsolete, forms; they were used by Dante and Petrarca. 2. Have, for ha, is used in poetry, and also in the familiar language of some provinces. 3. A-vé-mo, for abbiamo, is used in familiar language. Also Petrarca has it in his poetry. The ancients wrote, likewise, ag-giá-mo for abbiamo, but it is now quite obsolete. In Tuscany they also say a-viá-mo for abbiamo. 4. The terminations of the imperfect tenses of all Italian verbs in eva have a close e-thus, é-va; for example, fa-cé-va, I did; di-cé-va, I said, etc. Va is the termination of the first and third person, and vi of the second person singular in the imperfect tense of all conjugations; for example 1st Conj. A-má-va, a-má-vi, a-má-va, I loved, thou lovedst, he loved. 2nd Conj. Te-mé-va, te-mé-vi, te-mé-va, I feared, thou fearedst, he feared. 3rd Conj. Dor-mí-va, dor-mí-vi, dor-mí-va, I slept, thou sleptst, he slept. 5. A-vé-vi, for avevate, is a Florentinism, just like eri for cravate. Eb-bi-mo, for 6. Éb-ba-mo, for avemmo, is a Florentinism. avemmo, is in use, but not quite correct. 7. Eb-bo-no, for ebbero, was once much in use on account of euphony, and is a form similar to a-vrêb-bo-no for a-vrêb-be-ro. 8. Strictly speaking, a-ve-rd, a-ve-rá-i, a-ve-rà, a-ve-ré-mo, a-ve-ré-te, a-ve-rán-no, for avrò, etc., is the regular form of the future of avere, while avrò is a contraction. Averò is still in use among the people, and also was used by ancient writers. but must be considered as obsolete. A-rò, a-rá-i, a-rò, a-ré-mo, a-ré-te, a-rán-no, for avrò, etc., is still in use among the people of Florence, and was also sometimes used by ancient writers, but educated people ought neither to say nor to write it. The ancients said avroe for avrò, and avrae for avrà, which of course is now obsolete. 9. A-ve-rê-i, a-ve-ré-sti, etc., for avrei, avresti, etc., are the original forms of this tense, and must be considered as obso lete, though they are still in the mouths of the people. A-tr? is a contraction for avrei or avrebbe. 10. A-vrí-a-mo or a-vriê-mo, for avremmo, are poetical forms. A-vrés-si-mo is the usual Romanism, and a-vrêb-ba-mo the usual Florentinism, for avremmo. Both are, of course, erroneous. 11. A-vrić-no, for avrebbero, is a poetical form. III. REMARKS ON THE IMPERATIVE MOOD. 1. For non dê-vi a-vé-re, thou must not have, as explained before. 2. A'b-bi, for the third person singular (áb-bia, let him or her have), and áb-bil-no, or áb-bi-no, for áb-bia-no, are not quite correct, and perhaps somewhat vulgar; though, as a familiar form, the Tuscans frequently say abbino for abbiano, and even apply this form to all verbs of the second conjugation; and it must be added that the best writers of the sixteenth century used it. IV. REMARKS ON THE SUBJUNCTIVE MOOD. 1. See the preceding remark-of equal application here. 2. The imperfect tenses of the subjunctive mood, and of the second conjugation (to which avere belongs), ending in essi, always have a close e-thus, és-si; for example, te-més-si, I might fear; cre-des-si, I might believe; etc. 3. Avessi, thou mightest have (or with se-se tu avessi, if thou had), is exclusively of the subjunctive mood; while avesti, thou hadst, is exclusively of the indicative mood and of the indeterminate preterite. 4. In low style avessi is sometimes used for avesse. 5. In low style avessemo, for avessimo, has been used by old poets. 6. A-vés-so-no, in the place of avessero, for the sake of euphony, is a form which occasionally is applied to all verbs. A-vés-si-no, for avessero, is a Florentinism, like abbino. ADDITIONAL REMARKS. Both a-vé-re and ês-se-re are irregular verbs of the second conjugation, and the compound tenses of both these verbs are 1. Ab-biên-te for avente, ab-biú-to or aú-to for a-vi-to, and taken from themselves without the aid of another verb, with ab-bien-do for avendo, are obsolete forms. this difference, however, that essere has its past participle taken from the verb stá-re, which, by a long usage, has become its property, while the original past participle, essuto or suto, is quite obsolete. Essere is an intransitive verb, denoting being or existence; avere is a transitive or active verb, denoting possession of a thing, and capable of being employed in a passive form, which is not the case with essere, though this is the auxiliary, by the aid of which the passive form of all Italian verbs is conjugated. essere. In essere the participle stato changes with the gender and number of the person or thing to which it refers; as--o (0) sono stá-to, i-o (dôn-na) só-no stá-ta, nói (má-schi, males) siá-mo stá-ti, với (fém-mi-ne, females) siê-te stá-te, etc. This is also the case with all passive, intransitive, and reciprocal verbs, which are conjugated by means of the auxiliary In the compound tenses of avere, on the contrary, the participle avuto does not change its number and gender: for example—áo (uô-mo) hô a-vú to, tu (đôn na) hả-i a-vi-to, nói (d-schi) ab-bi-mo avu to, với (fêm-mi-ne) a-ve-tea-vi-to. This is also generally the case with the past participles of all active verbs which are conjugated by the aid of avere. Sometimes, however, the participles of active verbs must adopt the number and gender of the verb to which they refer, which will be explained later. For the negative form of the auxiliaries avere and essere, and indeed of all other verbs, non is used, and always placed before the verb; as C'è or v'è, there is. Ci sono or vi sono, there are. And so of the other tenses :- C'è or v'è ú-na gran quan-ti-tá, ti is a great quantity. Ci só-no or vi só-no dél-le per-só-ne, there are persons. Cé-ra ú-na vôl-ta un sá-vio Grê-co, there was once a wise Greek. V-ra-no de' po-po-li, there were nations. Note that ec-ci and ev-vi in the last sentence are written thus according to the rule that a monosyllable like è, in compositions, loses its accent and doubles the initial (unless an s impare) of the suffixed word. In similar cases avere (with vi before it) may be used for essere, and even stand in the singular, though the accompanying noun is in the plural; as Vha (for v' hanno) de' prín-ci-pi, | V' ha or háv-vi mól-ta gên-te pô-vethere are princes. ra, there are many poor people. ha mól-te co-se, there are many Mól-ti sol-dá-ti v' a-vé-a, there were things. many soldiers. Si dán-no di quelli che so-stên-gono -, there are some who maintain -. Dán-no-si qui de' gran com-mer-cian-ti ? are there great merchants here? The words ci and vi (here, there, in this or that place) merely being local adverbs, it is clear that they must be suppressed when speaking of time; as E un mé-se, só-no dú-e án-ni, it is a month, it is two years. Ciò ac-cád-de dú-e mé-si fa, this happened two months ago. KEY TO EXERCISES IN LESSONS IN ITALIAN.—XXII. EXERCISE 31. 1. Voi avete bel tempo per viaggiare. 2. Adesso abbiamo continuamente belle giornate. 3. Egli ebbe l' anno scorso un gran giardino fuor di città nel quale trovansi bei fiori e begli alberi da frutti. 4. Quel libro tratta della vita di Santo Stefano e di San Giorgio, ed in questo vi sono spiegazioni d' alcuni passi dalle epistole di San Paolo e di San Pietro. 5. Teodosio il Grande morì a Milano nelle braccia di Sant' Ambrogio. 6. Quello scritto contiene un bel pensiero sui vantaggi di commercio. 7. Quei principi sono felici i quali vengono amati de' sudditi. 8. In questo affare bisogna avere gran circospezione e gran coraggio, 9. Roma e Cartagine avevano tra di loro gran guerre. 10. Demostene era un grand' oratore greco. 11. Egli è un buon giovane, e ha una gran disposizione d' imparar tutto facilmente. 12. Le gemme sono corpi diafini; tali sono i diamanti bianchi, il rubino rosso, il zaffiro turchino, lo smeraldo verde, ed il giacinto giallo. 13. Le perle, piccole o grande, crescono in conchiglie, ed i coralli in mare, nella forma di arboscelli. 14. Lo zio mi ha donato un libro francese. 15. Goffredo ha una gran provigione di vino ungherese ed austriaco. 16. I cavalli spagnuoli sono così cari come l' inglese. LESSONS IN ALGEBRA.-XXIX. A root whose value cannot be exactly expressed in numbers is called a SURD, or irrational quantity. Thus, 2 is a surd, because the square root of 2 cannot be expressed in numbers with perfect exactness. In decimals, it is 1-41421356 nearly. Every quantity which is not a surd is said to be rational. By RADICAL QUANTITIES is meant all quantities which are found under the radical sign, or which have a fractional index. REDUCTION OF RADICAL QUANTITIES. CASE I. To reduce a rational quantity to the form of a radical without altering its value. Raise the quantity to a power of the same name as the given root, and then apply the corresponding radical sign or index. EXAMPLE.-Reduce a to the form of the nth root. The nth power of a is a". Over this place the radical sign, and it becomes "a". It is thus reduced to the form of a radical quantity, without any alteration of its value. For" an = a11 = a. n N.B. In cases of this kind, where a power is to be reduced to the form of the nth root, it must be raised to the nth power, not of the given letter, but of the power of the letter. Thus, in the fifth example, Exercise 48, a is the cube, not of a, but of a2. CASE II. To reduce quantities which have different indices When there is, there are, and similar phrases, have the to others of the same value having a common index. CASE IV. To reduce a radical quantity to its most simple terms; i.e., to remove a factor from under the radical sign. Resolve the quantity into two factors, one of which is an exact power of the same name with the root. Find the root of this power, and prefix it to the other factor, with the radical sign between them. This rule is founded on the principle that the root of the product of two factors is equal to the product of their roots. It will generally be best to resolve the radical quantity into such factors, that one of them shall be the greatest power which will divide the quantity without a remainder. N.B.-If there is no eract power which will divide the quantity, the deduction cannot be made. EXAMPLES. Remove a factor from 8. n nan, or an; and a Reduce a (x – b) to the form of a radical. a(x − b) 3 — 3⁄4/a3(x − b) = = EXERCISE 49. 1. Reduce 18 to its simplest form. 2. Reduce 3/6¥3c. 3. Reduce 4. Reduce ab. 5. Reduce (a3-a2i)1. 6. Reduce (54)}. 7. Reduce 98a2x. 3 8. Reduce a3 + a3b2. 9. Reduce 2ab (2aba) }. 10. Reduce() 11. Reduce 2/2. 12. Reduce 463 √c. 13. Reduce 5/6 to a simple radical form. 14. Reduce √5a to radical form. 15. Reduce 5 and 6 to others with the common index . 16. Reduce a2 and a to others with the common index. 17. Reduce 98 to its simplest form. 18. Reduce √243 to its simplest form. 19. Reduce 3/54 to its simplest form. 20. Reduce 780 to its simplest form. 21. Reduce 9381 to its simplest form. 22. Reduce √x+ar to its simplest form. 23. Reduce 198a2x to its simplest form. √x3 a simple 24. Reduce simplest form. ADDITION OF RADICAL QUANTITIES. a2x2 to its It may be proper to remark, that the rules for addition, subtraction, multiplication, and division of radical quantities depend on the same principles, and are expressed in nearly the same language, as those for addition, subtraction, multiplica. tion, and division of powers. So also the rules for involution and evolution of radicals are similar to those for involution and evolution of powers. Hence, if the learner has made himself thoroughly acquainted with the principles and operations relating to powers, he has substantially acquired those pertain ing to radical quantities, and will find no difficulty in understanding and applying them. When radical quantities have the same radical part, and are under the same radical sign or inder, they are like quantities. Hence their rational parts or coefficients may be added in the same manner as rational quantities, and the sum prefixed to the radical part. If the radical parts, after reduction, are different, different exponents, then the quantities, being unlike, can be The greatest square which will divide 8 is 4. We may then added only by writing them one after the other with their resolve 8 into the factors 4 and 2; for 4 X 2 = 8. signs. |