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LESSONS IN PENMANSHIP.-IX.
On inspecting any copy-slip that has the letter 1 in it, it will
be found that the letter p extends as far below the line b b as THE letter p is the first letter that the learner has met with the letter l extends above the line a a. That portion of the that extends below the line b b, and it will be necessary here to letter p which extends above the line a a is longer by one-sixsay something about its proportions, as they are given in Copy teenth of an inch than the distance to which the letter t extends slips Nos. 28, 30, and 31.
above the same line, or the distance between the top of the It will be remembered that in "large text," the distance bottom-turn of the letter i and the dot above it, as may be seen between the lines a a, b b, that contain what we have called the by examining Copy-slips Nos. 30 and 31. body of the letter, is, or ought to be, exactly half an inch; and We have been thus particular in dwelling upon the distances as the line c c is midway between the lines a a, b b, the distance to which letters such as t, 1, h, p should extend above a a, or j
7 hut full um
between each of these lines and the central line, cc, is a quarter below bb, in order to induce the learner to pay strict attention of an inch. Now the distance between a a and the line ff, at to the relative proportions of his letters. The importance of which the long straight stroke of the letter p is commenced, is this will be seen by any one who is curious enough to extend also a quarter of an inch, and is equal to the distance between these letters to a greater or less length above a a, or below b b, da and c C, or between b b and c c. The distance between 6 6 than is allotted for their extension in our Copy-slips. The and 9 9, the line at which the long straight stroke terminates, is general appearance of handwriting that would otherwise be rather leas than half an inch, or, to speak in exact terms, just good, is often completely spoiled by a want of proper proportion seven-sixteenths of an inch; that is to say, if an inch were in the heads, loops, and tails of the letters. Those who wish to divided into sixteen equal parts, the distance between bb and ggbe distinguished for writing a plain and legible hand, must aim is equal to seven of them, while the distances fa, ac, cb, on the at the neatness and beauty of the writing that is found in old straight line f g, are each equal to four-sixteenths of an inch, deeds, and books copied by the monks who lived before the which is merely another expression for a quarter of an inch, as time of Caxton. The letters of these famous penmen are as our learners will find when they have got on far enough in regular in their proportions and as sharply and delicately deArithmetic to be working at Vulgar Fractions.
fined as if they had been carefully printed from well-cut type.
LESSONS IN ARITHMETIC.-IX.
This method of arrangement evidently gives the greatest
number of times which each prime factor occurs in any one of LEAST COMMON MULTIPLE.
the given numbers. Thus 2 occurs three times in 72, 3 occurs 1. One number is called a multiple of another when it can be twice in 72, and 7 occurs only once-viz., in 42 and 84. divided by the latter without a remainder. Thus, a measure and a multiple are the converse of each other.
EXERCISE 21. If a number divides another without remainder, it is said to be 1. Find the least common multiple of the following numbers: a measure of it, and the latter number is said to be a multiple
1. 15 and 45.
13, 1, 2, 3, 4, 5, 6, 7, 8, and 9. of the first.
2. 63 and 18.
14. 657, 350, 876, 1095, 2190, A common multiple of two or more numbers is a number 3. 6, 9, and 15.
and 5795. which can be divided by cach of them without a remainder. 4, 8, 16, 18, and 24.
15. 42, 12, 84, and 72. It will clearly be a composite number, of which each of the given 5. 9, 15, 12, 6, and 5.
16. 9, 12, 72, 36, and 144, numbers must be a factor, for it could not otherwise be divided
6. 5, 10,
17. 8, 12, 20, 24, and 25. by them.
7. 24, 16, 18, and 20.
18. 63, 12, 84, and 7. The same numbers may clearly have an infinite number of
8. 36, 25, 60, 72, and 35.
19, 54, 81, 63, and 14, 9. 27, 54, 81, 14, and 63.
20. 75, 120, and 300, common multiples, for any one common multiple having been
10. 72, 120, 180, 24, and 36. 21. 96, 144, and 720. found, another may be obtained by multiplying it by any
ll, 375, 850, 3200, and 5085. 22, 256, 512, and 1728, number.
12. 7, 11, 13, and 5.
23. 375, 550, and 3400. The continued product of two or more numbers will always give a common multiple of those numbers. The least common multiple of two or more numbers is the
LESSONS IN GERMAN.-VIII. least number which can be divided by each of them without a remainder.
SECTION XVI.-USE OF THE DEFINITE ARTICLE;
PROPER NAMES, ETC ETC. Thus, 70 is the least common multiple of 2, 5, and 35.
2. The least common multiple of two or more numbers is The plural of Minn is Männer ; except in compounds, where it evidently composed of the continued product of all the different is generally leute (S XV. Note), as Cantmann, countryman : Landprime factors which compose the given numbers, each one being leute, country-people. Zimmermann carpenter; Zimmerleute, car. repeated as often as the greatest number of times it occurs in penters. Hauptmann, captain ; Hauptlcuie, captains. Kaufmann, any one of the numbers. For if it did not contain all the prime merchant; Kaufleute, merchants. factors of any one of the numbers, it could not be divided by Volt corresponds mainly to our word people. Unlike this, that number.
however, it has different forms for the two numbers, as :-Die On the other hand, if any prime factor were employed more Franzoien fint ein lebhaftes Volt; the French are a lively people. times than it is repeated in any one of the given numbers, it Die Jürsten schwelgen, und das Volt leitet ; the princes revel, and the 'would not be the least common multiple.
people suffer. Alle Völfer auf Erten, 1 Moses xviii. 18; all the For the sake of brevity the words “ least common multiple" | nations of the earth, Genesis xviii. 18. are sometimes written L. C. M.
The word one, as a pronoun, is, in English, often inserted 3. EXAMPLE.Find the L. C. M. of 12, 126, and 735.
after an adjective, to avoid the repetition of the noun; in Ger. These are respectively equal to
man, however, the adjective in such a case stands alone, as :2 X 2 X 3, 2 X 3 X 3 X 7, 3 x 5 x 7 x 7.
Er hat einen guten Hut, und ich habe cinen schlechten; he has a good hat, Now 2, 3, 5, 7 are all the different prime factors which occur and I have a bad (one). Ich habe gute Hüte, und er þat schlechte; I in any of the numbers; and the greatest number of times which have good hats, and he has bad (ones). Or hat guten Wein, und ich 2 occurs is twice-namely, in the first; the greatest number sabe schlechten : he has good wine, and I have bad (wine). which 3 occurs is twice--namely, in the second ; 5 only occurs The adjective and participle preceded by an article are often once-namely, in the third ; and the greatest number of times used substantively, as well in the singular as in the plural, as :which 7 occurs is twice-namely, in the third. Hence the Der Zufrietene (Sect. IX. 2) ist glüdlich; the contented (man) is L. C. M. required will be
happy. Die Zufriedene ist glüdlich; the contented (woman) is 2 X 2 X 3 X 3 X 5 X 7 X 7; that is, 8820.
happy. Die Zufriedenen find glüdlich : the contented are happy. 4. The process, then, of finding the least common multiple of Der Sterbente, tie Sterbente; the dying (man), the dying (woman).
Ein Zufriedener (Sect. X.) ist glüdlich; a contented (man) is happy. two or more numbers is reduced to that of splitting up the Die Lebenten; the living. qumbers into their prime factors. This may be effected, however, by a more convenient method article, converted into abstract nouns, as :
1. Adjectives in German are often, by means of the definite of arrangement than splitting each number separately into he adores the beautiful.
-Er verehrt tas Schöne ; factors would be, for which we give the following
2. The use of the definite article before nouns, taken in & Rule for finding the least common multiple of two or more general sense, is much more frequent than in English, as :
:- Der numbers. Write down tha numbers in a straight line apart from each diamond is a precious stone. Dag Golt ist ein edeld Metall ; (the)
Tiger oft flink : the tiger is agile. Der Diamant ist ein Edelstein ; the other. Divide by the least number which is a measure of two or more of them, and set down the quotients and the undivided gold is a precious metal. Die Luft ist ein Element; the air is an
element. pumbers in a liné below. Take again the least number which Die Seele ist unsterblich : the soul is immortal.
Das Wasser ist ein Element: (the) water is an element. is a measure of two or more of these numbers last set down, and sterblich); (the) man is mortal. Die Faulheit ist ein laster; (the) idle
Der Mensch ift perform the same operation as before. Continue it until there
ness is a vice. are no two numbers which can be divided by any number greater
The plural is used in the same manner, as :-Die Tiger fint flink; than unity. The continued product of all the divisors, and the numbers set down in the last line, will be the least common
(the) tigers are agile.
3. The definite article is sometimes used instead of the posmultiple required. 5. EXAMPLE.—To find the L. C. M. of 12, 42, 72, and 84.
sessive pronouns, as :-Er hat ein Buch in der Hand; he has a The process will be sufficiently understood from the following with the (its) father.
book in the (his) hand. Das Kind ist bei dem Vater ; the child is working :
4. Proper names and titles are often preceded by the defi
. 2) 12, 42, 72, 84
nite article, as :--
--Wo ist der Scinrich? where is (the) Henry? Det 2) 6, 21, 36, 42
Kaiser sicinrichy, the Emperor Henry. Der König Heinrich ; (the) King
Henry. 3) 3, 21, 18, 21
The definite article likewise commonly precedes the adjective
qualifying a proper name, as :-Die idòne Helene; the beautiful 7) 1, 7, 6, 7
Helen. Der arme Richard ; (the) poor Richard.
The article is also generally used before the word Sdule, 1, 1, 6, 1
Kirche, Marft, Müble, xe., as :-Gr ist in der Schule: he is (in) at (the) Hence the L. C. M. is 6 X 7 X 3 X 2 X 2; that is 504. school. Gr ist in ter Kirche ; he is (in) at (the) church. Er ist auf
tem Markt; he is at the market. Er ist in der Mühle ; he is in the Der Herr Gesand'te ist so eben mit The (Mr.) Ambassador, with his mill. Er geht nach der Mühle; he is going to (the) mill.
seiner Frau Gemahlin ab'gereisi lady consort, has just now 5. The word Herr, when placed before a proper name, answers
departed. to our Mr., as :- -3it Herr N. hier ? is Mr. N. here ? frau in Nehmen Sie auf dem Sopha gefál. Tako (Fou) seats upon the sofa, the like position signifies Mrs., as :-Wo ist Frau N. ? where is lijst Plag, meine Damen, meine if you please, (my) ladies, Mrs. N. ? fräulein, thus placed, answers to our word Miss, Fraulein, oder meine Herren. (my) young ladies, or (my) 25 : -riulein N. ist hier ; Miss N. is here. Guten Morgen, Herr N.,
gentlemen. frau N., gräulein N.; good morning, Mr. N., Mrs. N., Miss N. Frau N., ihre Friulein Tochter, und Mrs. N., her (Miss) daughter, Instead of frau the French Madam is often used, as :- -Matam ihr Herr Sohn sind in Ihrem and her (Mr.) son
are in N.; Mrs. N.
your room. In address, when the name is omitted, the possessive pronoun
EXERCISE 21. precedes the words Herr and Fräulein, as :-Guten Morgen, mein Hert, mein Fräulein; good morning, Sir, Miss.
1. Die Zimmerleute, Schreiner, Schneider, und Maurer find Hantwerfer.
3. Dic Engländer sind In the plural the form of address is: Meine Herren! Gentlemen! 2. Die Vergleute arbeiten in der Tiefe der Grre. Meine Damen! Ladies! Meine Fräulein! Young ladies !
ein ruhiges Volf. 4. Fleißige Hantwerksleute verdienen in America viel The word Fräulein, when connected with the name, is used like Oelt. 5. Der Reiche hat einen guten Rock, und der Bettler cinen schlechten. its corresponding word in English, as :—Sird die Fräulein N. zu 6. Der Tugenthafte scheut das Lafter. 7. Der Glüdliche bedauert den UnGauje? Are the Misses N. at home?
glüdlichen. 8. Die Ungelehrte beneidet die Gelehrte. 9. Das Gelehrte ist In ceremonious address the word Berr is prefixed to titles, nicht immer das Nüßliche. 10. Herr N. ist in dem Zimmer. 11. Frau
12. Ich gehe mit Ihnen, mein Herr.
13. Ich 29 :—Herr Pristent; Mr. President. Herr Sprecher; Mr. Speaker. N. ist in dem Theater.
14. Ich war gestern Gert Pastor ; (Mr.) Pastor. Herr Oberst; (Mr.) Colonel. Gerr Pro- wünsche Ihnen einen guten Morgen, mein Fräulein. fepice; (Mr.) Professor. Herr Lehrer ; (Mr.) Teacher. Herr Ritter ; die Frau Gesanttin nicht. 17. Der Herr Minister ist ebenfalls dort.
in einer Damengesellschaft. 15. Frau N. ist sehr munter. 16. Id febe
18. Sir Knight.
19. Guten Abend, Herr Professor. 20. Harr
, preceded by the definite article, is applied to these titles Die Frau Hufräthin hat Trauer. as well in the third person as in the second, as :-Wo ist der Werr Wo ist Ihre Frau Gemahlin, Ihr Herr Sohn, und Ihre Fräulein Tochter ? Prijtent? Where is the (Mr.) President ? Wissen Sie
, wo der 21. Sie sind in tem Concert. 22. Vom (§ 4. 2) Erhabenen zum Lächer. Gerr Oberst ist? Do you know where the (Mr.) Colonel is ?
lichen ist nur ein Schritt. 23. Der Reiche hat zwei* Häuser, drei Knechte, OBS.—The words in parentheses are the literal translations of vier Pferde, zwölf Odysen, und ac)tzig Schafe. 24. Dieser Jäger hat fünf the German ; they are given that the pupil may clearly perceive Hunde, und jener hat acht..
EXERCISE 22. the different modes of expression of the two languages. Peculiarities of this kind are called “the idiom of a language," and 1. The Germans are a diligent people. 2. My father knows the differences " differences of idioin."
[fennt] a learned professor, but an unfortunato (one). 3. The The word frau is prefixed to titles or appellations of women, unlearned [lingclehrte] avoids the learned. 4. The dying (man) ag :- Frau Vonnerin ; Lady Patroness. Frau Acbtissin; Lady Ab- has a book (Buch) in his hand. 5. He adores the beautiful and bess. Frau Gemahlin ; Lady Consort.
the sublime. 6. Gold is a precious metal, silver is likewise, but These words are also prefixed to designations of relationship, copper (Kupfer) and iron are [sino] not. 7. Havo you seen poor 23:—Wo ift 3hr Herr Vater? Where is your (Mr.) father? Sein Henry and little Helen ? 8. Your friend, the captain,t was in Ser Bruter ist bier. His (Mr.) brother is here. Ist Ihre Frau the theatre. 9. I see the colonel yonder in the garden. 10. Mutter zu Hause? Is your (Mrs.) mother at home ? Ihre Fräulein | Good morning, president. 11. Is your mother at [gu] home Sorefter war da. Your (Miss) sister was there.
[Hause] ? 12. Good [guten] evening [Abend], Sir; where is your VOCABULARY.
sister and your brother ? Utend, m. evening, Gestern, yesterday. Schauen, to view, arbeiten, to work. Hand'werfer, m. me- Scheuen, to shun. Betau'ern, to pity. chanic.
Schlecht, adj. bad,
LESSONS IN DRAWING.-V. Benei'ten, to envy.
Herr, m. Mr., Sir, base. Bergmann, m. miner. Lord.
Schreiner, m. joiner.
We cannot urgo too strongly on our pupils the necessity of Bettler, m. beggar. Hofrathin, f. wife of Schritt,m.step, stride. going to work carefully and deliberately. Consider well what Concert', n. concert.
counsellor of Seben, to see, per. you have to do before you begin. Endeavour to make no line or De'mengesellschaft, f. the court (Sect. ceive.
touch that is not to the purpose. If you cannot satisfy yourself society of ladies.
on the first trial, be not disappointed, but try again, and again. (See Sect. VIII. 1.) Läch'erlich, adj. ludi- Tiefe, f. depth.
Recall to mind the errors you made in the first attempts, which Dort, there, yonder.
crous, ridiculous. Trauer, f. mourning, you should keep by you, that you may often refer to them. In G'benfalls, likewise. Laster, n. vice.
your next trial you will do better. You will have advanced a
sorrow. Groe, f. earth. Madam'
, f. Mrs., ma-Tu'gendhaft
, virtuous. certain step, and onward will be your progress, as surely as you Grha'ben, sublime. dam.
Un'gelehrt, unlearned, persevere. Never fatigue yourself over your drawing. The Grau, J. Mrs., woman, Maurer, m. mason. illiterate.
moment you work without a will it should be laid aside. wife. Minister, m. minister. Un'glüdlich, unhappy,
And now—and this is a point of the greatest importance to the Gelehrt', adj. learned. Mit, with.
learner--we must further urge on those who are working with Gemat'lin , f. consort, Morgen, m. morning. Dertie’nen, to earn, position when drawing. In Fig. 35 the proper position of the
us by means of these lessons, to endeavour to acquire a good wife.
Muster, n. sample. deserve. Gesantr'in,f.ambassa- Munter, adj. lively. Viel, adj. and adv. paper on which you are drawing, and the copy which you are dress. Nüßlich, adj. useful.
endeavouring to imitate, is clearly shown. Your paper should Glüdlit, adj. happy, Professor, m. profes- Wūnichen, to wish, about two or three inches in a foot, or on a flat and perfectly
be placed before you on a desk, with a slight inclination of fortunate.
level piece of board, to which it may be secured by flat-headed RÉSUMÉ OF EXAMPLES.
drawing-pins, and which may be supported at the proper incliDie Hüt'tenleute haben schwere Ar'. The furnace-men have severe nation on a book or a piece of wood, as at b. The drawing beiten
which you are copying should be supported as at a, by a light Die meisten Volfer A'fiens haben (The) most nations of Asia have easel or frame with a leg, fastened by a hinge to the upper noch Wißen.
part of it, by which tho inclination of the frame may be reguDer Water bat einen schwarzen Hut, The father has a black hat, and lated at pleasure. und der Soon cinen tveißen. the son a white (one).
The position in which you sit should be perfectly easy, and Das Sdöne ist lieblich, aber nur das The beautiful is lovely, but in no way painful to the chest. There is no necessity for lean. Gute adh'tungswerth.
only the good (is) worthy of
* Zwei, two; trei, three; vier, four ; zwölf, twelve; achtzig, eighty;
fünf, five; acht, eight. Madam is the same as the French Madame (my lady), but never † Remember that in German " the captain" must be rendered "the spelt with the e as is the French word.
Mr. Captain," etc.
ing over your work in an ungraceful or painful attitude. The longer whilst we make a few remarks upon some of the pecusye should be as nearly as possible directly opposite the centre liarities of Angular Perspective. No doubt it is much more of your drawing, and the inclination of your paper and copy difficult to understand than parallel perspective, arising from should be such that a line passing from your eye to either paper the great variety of positions in which objects may be placed or copy, when you are at work, should pass through the centre for if the lines are, on the one hand, ever so slightly out of the of the copy at a in Fig. 35, or the centre of the drawing at b, as perpendicular from the picture plane, or, on the other, in the nearly as possible at right angles to their respective planes. least degree vary from the parallel to the picture plane, the
It is unnecessary to give directions as to the manner of holding treatment necessarily comes under the rules of Angular Per. your pencil. Your own judg.
spective. Let us try to make ment must direct you in a great
this clear by the help of Figs. measure as to that. It matters
36 and 37. little, so that you feel the in.
Fig. 36 is a case of parallel strument fit your fingers easily.
perspective (see p. 72, Fig. 27a). If proper attention has been be.
Fig. 37 has its sides a b and stowed upon the primary instruc
e f slightly removed from the tions that we have given, you
perpendicular c d, and the sides have already learned the im.
a e and b s in the same proportance of depending not solely
portion removed from the line on your fingers, but also on the
e g, parallel to PP; consequently action of the wrist and arm.
it presents the angle at e to the The hand should not be suf
picture plane. (We shall prefered to rest upon the paper upon
sently be under the necessity which you are drawing, if it can
of seeking a little help from Geobe avoided, but have a spare
metrical Perspective, in order piece to lay under it while you
to make ourselves more clearly are at work. It will serve
understood.) There is another another purpose — to try the
peculiarity in connection with points of your pencils upon, or
this relating to the position of the points of your pens, crayons,
the vanishing points; we request and brushes when you are sufficiently advanced to draw with the pupil to look at Fig. 38: he will observe that the angle of the pen and ink, or to paint in water-colours. Begin at once to pro- building, a b, is nearest the eye, whilst the side a b c d retires gerve your drawings in a portfolio. Even when you have failed one way to v P 1, and the side a bef retires another way to m many attempts you should keep them by you. Destroy nothing v P 2. Now, when he sees this he will probably say, “Yes, that you do, and you will soon learn to do nothing that you these sides certainly do retire as so stated, but I should like to would desire to destroy. Preserve order in the disposition of be informed why these two vanishing points are placed where all your materials : much time and vexation may be saved by they are. Is there any rule for so placing them ? or is it Fig. 40. Fig. 41. d!
VP3 Fig. 36.
Fig. 38 it; and, above all things, remember that what is worth doing, merely a matter of choice ?-in short, can I place them anywhere is worth doing well.
I please ?” These are very fair questions, and we will en. We propose now to give some instructions in Angular Per. deavour to answer them. Of course, the house (Fig. 38) must spective; we use this term when the object presents an angle, have a ground-plan, which will be placed with regard to the picand not a side, to the picture plane, that is, when the angle is ture plane as it is shown in Fig. 39, the angle towards us, and nearest as and all the sides retire ; this occurs especially when the sides retiring. Now let us suppose we are standing at s P all rectangular forms, such as buildings, boxes, and things of a (station point), from which place we are to make our drawing : similar shape and character, are so arranged. (See p. 72, Fig. from this place we determine our vanishing points, and the 276.) Before we proceed to explain the method of drawing distance these vanishing points are apart will determine whether objects so placed before us, we must detain the pupil a little I we are near or at a greater distance from the object. Then to
determine onr vanishing points, we must give the following geo- points: for instance, let him trace out the lines a e and b fin metrical rule :-“Draw a line from the station point, parallel to Fig. 38, they will meet at v p 2; and the lines on the other the ground plan as far as the plane of the picture, from which draw side, a c and b d, will meet at v pl. In an engraving, the & perpendicular line to the horizontal line (line of sight); this vanishing points for all horizontal retiring lines may be found will give the vanishing point.” Let as look at Fig. 39, we shall in this way, and they will determine also the line of sight which find that the line a b is drawn from s pl to the picture runs through these points. If he discover that these horizontal plane, parallel to one of the retiring lines of the ground plan, retiring lines do not meet in the same point, it will be because cd, which gives VP1; also a e is drawn parallel to fc, the they are not parallel retiring lines; that is, the objects themother retiring line of the ground plan which gives us v p 2. selves are not placed in a parallel position with each other. This But if the station point had been further off, as at 8 p 2, the leads to another observation connected with this last remark; if line a b would have been from s p 2 to h; therefore at h there are fifty retiring lines, and all parallel, there will be only would then be found v p1; so on the other side the v p 2 one vanishing point for them all; but if amongst these fifty would have been at i. Suppose the station point were placed at there are not two parallel, there will be fifty vanishing points. SP3, then the vanishing points would be nearer each other. As the pupil, we hope, will clearly comprehend this interesting So it will be seen, the further the vanishing points are apart, feature in perspective drawing, he may apply the rule when he the further we are from the object; and the nearer we are to the has an engraving before him. We know that the rectangular object the nearer together are the vanishing points. Our objects ) tops and bottoms of windows and doors are horizontal, and
then, in giving this little explanation, is to account for placing parallel with the eaves and horizontal ridges of roofs, the courses the vanishing points. To carry this rule out by producing a of the bricks, etc. Let him trace out as many of these lines drawing of the elevation of the house from the ground plan, will as he can, if he understands they are intended, as in the object be considered hereafter.
itself, to go off in the same direction, and he will find them meet By this explanation we only undertake to satisfy our pupils at the same vanishing point, and soon discover whether the that we can make a correct drawing of the building if the engraving is correct or not in the grammar. The uneducated eye vanishing point was at h, b, or k; only observe, that if we may not detect small faults in the general appearance of the approached too near, the angle of sight, m, would be too large, engraving, and thousands of drawings and paintings by really Bo much so as probably to become as great, or greater than 60o. clever artists pass muster, and are admired, although they may (See p. 72.) While writing these remarks on Angular Per. be full of mistakes; just as in speaking, the grammatical errors spective, we found that it was absolutely necessary to give these habitually made by uneducated men are not even known to be geometrical reasons for the positions of the vanishing points, such among themselves, but an educated man will notice them, because as many lines in a picture retire and vanish elsewhere although he may not remark upon them. After the pupil has than at the point of sight, we felt bound to give these reasons, discovered the vanishing point for the horizontal retiring lines which need not cause the pupil to imagine there is anything to in the engraving, he will then have found the position of the discourage him, as the mode of finding them in a picture, as well line of sight; then, in making his drawing, he must begin by as when drawing from Nature, is very simple. Here, then, the placing this v p on his paper, and proceed by marking in pupil may ask, “ If I have a drawing before me to copy, in which the nearest line to the vanishing point, and so on, line after the vanishing points are not marked, how shall I find them ?" line, as we have before said. We know from experience the Let us suppose the copy is an engraving (and the vanishing great advantage of this method, and have frequently remarked points are never shown in engravings), let him trace ont the the rapid progress that has been made by those who have feared retiring lines in the picture—we think there can be no difficulty that drawing was an art too difficult for them to attain. in recognising them—these lines traced out will give the vanishing The method of drawing Fig. 38 will be as follows:-Draw the