Abbildungen der Seite

that the space thus assumed is not taken too long in proportion which the reader may construct a projection for a map of to the base line of the map, and set off from E along the straight Europe, taking the dotted line A B to represent the distances line E B eight of these spaces, and number the points, begin to be set off along the arc representing the parallel of 45°; but ning with E, 35, 40, 45, 50, 55, 60, 65, 70, 75, as in the figure, it will be better for him to construct scales for himself, much and through the point marked 75 draw the straight line g h larger in size than the largest which we have given in Fig. 16. parallel to D. The line o I will be the limit of the map to- Distances equal to the line a b that represents 42:43 on the wards the top, and the points numbered upwards in succession small scale in Fig. 15, must now be set off on either side of the from E will be the points through which will pass the parallels central meridian, represented by the straight line A B, along the of latitude corresponding with the numbers. To find the centre arc O P Q, and straight lines must be drawn from F with a fine from which to draw the parallels of latitude, measure four drawing-pen through the points thus obtained. The dotted lines spaces upwards from the point marked 75, along A B, and num- between F and the top of the map need not be drawn by the ber them 80, 85, 90, and F. The point F, just five degrees learner. The remaining arcs must then be drawn with a compass higher than the pole marked 90, is the centre from which the pen, and the limits of the map to the east and west determined parallels of latitude are to be described. This point corre- by drawing the straight lines K L M N at right angles to the sponds to the point m in Fig. 14, in which the sphere is sup- base line CD, the former a little to the left of the meridian posed to be entered by a cone passing through it in parallels 5° west longitude, and the latter a little to the right of the 45 and 65, the spaces from 45 to M, along the line L m, in Fig. meridian 45o east longitude. The border lines should then be 14, and the spaces from 45 or p to F, along the line A B, in Fig. drawn as shown in the engraving. The double lines at the sides 17, corresponding in number, as the reader may ascertain on and top and bottom of the inner space, which contains the map, comparing them. The actual distance of the point F beyond should be divided into single degrees and ruled, as in the figure, the pole is 4° 30'30ʻ, when determined by calculations involving to present a distinction of colour, and thus afford a ready moans a knowledge of trigonometry; but for maps on a small scale of counting and measuring degree lines not marked and numit is near enough to consider it as being equal to five degrees, bered on the map. The meridians should be numbered in the and therefore to the space assumed to represent five degrees in border at the top and bottom, and the parallels of latitude at the construction of the projection. The point f being thus the sides. The Arctic Circle must be inserted in the form of a determined, the parallels of latitude may be described through dotted line at the distance of 1° 30' above the parallel of 65o. the points marked on the central meridian with a finely pointed A blank space should be left in the upper left-hand corner, or pencil, or, as there is no absolute occasion to describe these arcs the lower right-hand corner, for the title and scale of geographiuntil the limits of the map on either side are determined, it will cal and British miles. To construct these scales, it must be be sufficient to draw a single arc O P Q through the point remembered that 60 geographical miles are equal to 69.07 marked 45 or P. This are should be drawn for some distance British miles, or that the line which was at first assumed as beyond the ultimate limits of the map on either side.

being equal to 5 degrees, represents 60 5, or 300 geographical We now proceed to draw the meridians, and to determine their miles, and 69.07 x 5, or 345 British miles, very nearly. position. The learner will have to refer to the table at the end In order to fix the po tion of places with accuracy, the student of this lesson, which shows the number of geographical miles is advised to divide the field of his map by pencil lines into contained in a degree of longitude under each parallel of lati- spaces of a degree each way, as shown in the lower part of tude, supposing the earth to be a perfect sphere in form. Fig. 17. This, however, can only be done when the map is

It will be remembered that the cone on which our map of on a sufficiently large scale. Learners are cautioned to use Europe is projected was supposed to pass through the sphere in Indian ink instead of common ink in drawing maps, as the ordithe parallels of 45° and 65°, and that the measurements on the nary ink will run and spoil the map where a final wash of colour cone along these lines are exactly equal to measurements along is given to the sea, and the boundary lines are distinguished by the same lines on the sphere, or, in other words, that the degrees contrasting tints. of longitude on these parallels, both on the cone and the sphere, TABLE SHOWING THE NUMBER OF GEOGRAPHICAL MILES IN are exactly equal. On looking at the table, we find that a

A DEGREE OF LONGITUDE UNDER EACH PARALLEL OF degree of longitude under the parallel of 45° is equal to 42:43

LATITUDE, THE EARTH BEING SUPPOSED TO BE A PERFECT geographical miles on the sphere, while a degree of longitude under the parallel of 65° is equal to 25:36 geographical miles. What we want to do, then, is to find a line bearing the same



Geog. Eng Geog. Eng. Geog. Eng. proportion to the line which we assumed at first to represent 5.

Miles. Miles. Miles. Miles. Miles. Miles.


Miles. Miles. degrees, as 42.43 geographical miles bears to 60 geographical miles, to enable us to set off points along the arc O P Q on

0 60.00 69.07 23 55 23 03:51 46 41.68 47.93 69 21.50 24.73 either side of the central meridian, through which the other

1 59 9969-06 24 54.81 63.03 47 40.92 47.06 70 20.52 23.60 meridians may be drawn from the point F. It would do equally 2 59.96 69-03 25 54-3862:53 48 40 15 46.16 71 19:53 22-47 well to find a line bearing the same proportion to the line 3 59.92 68.97 26 53-9362.02 49 39 36 45 26 72 18:54 21.32 assumed to represent 5 degrees as 25:36 bears to 60, and to set 4 59.85 68.90 27 59.46 6148 503857 44-35 73 17:54 20:17 off spaces equal to this line on either side of the central meridian 5 59 77 68.81 || 28 52-98 60.93 51 | 37 76 43:42 74 16:54 19-02

6 59 67 | 68.62 29 52 48 60-35 52 36.94 42:48 || 75 15.53 17.86 along the arc representing the parallel of 65°; but it is always

7 59 55 68.48 30 51.96 5975 53 36 11 41.53 76 14:51 16.70 safer, and ensures a higher degree of accuracy, to deal with the

8 59-4268:31 31 51.43 59 13 54 35.27 | 40 56 77 13:50 15-52 larger arcs and spaces instead of the smaller.

9 59 26 68.15 32 50 88 58:51 55 34:41 39 5878 12:47 14:35 To enable us to find lines bearing the required proportions to

10 59.09 67.95 33 50 32 57.87 56 33.55 39.58 79 11:45 13 17 the line originally assumed to represent 5 degrees, we must take 11 58.906773 34 49 74 5720 57 32 69 37:58 | 80 | 10:42 11.98 & straight line exactly equal to it, as in Fig. 15, and on it con- 12 58.69 67.48 35 49.15 56.51 58 31.79 36:57 81 9.39 10.79 struct a square. The sides of this square must be divided into 13 58 46 67.21 36 48:54 | 55.81 59 | 30 90 35.5+ 82 8:35 9.59 six equal parts, and numbered upwards at the points of section

14 58.22 66.95 37 47.92 55-10 60 30.00 34:50 83


841 from 0 to 60, while the top and bottom must be divided into ten

15 57.96 66.65 38 47.28 54:37 61 29-0933.45 8+ 6•27 7.21

16 57-6866-31 39 46 63 53.62 equal parts, the points of section between the extremities being

62 28:17 32:40

5.23 6.00 numbered from 1 to 9. Lines must then be drawn diagonally

17 57.38 65.98 40 45.96 52 85 63 27.21 31:33 4.18 4.81 18 57.06 65.62 41 45.28 52:07 64 26-30 30 24 87

3.61 across the square, from 0 on the left hand to 10 on the right

19 56.73 65.24 42 44 59 51.27 65 25:36 | 29.15

2:09 2 41 hand, etc., and perpendicular lines parallel to the sides through

20 56 38 6484 43 4388 50:46 66 21:40 28.06 89 1.05 1.21 the points of section numbered 1, 2, 3, etc. This diagonal scale, 21 56.01 64:42 44 43:16 49 63 67 23:4 26 96 90 0-00 0.00 constructed on the same principle as the scale shown in Lessons 22 55 63 63.97 45 42 43 48.78 68 22-18 25.85 in Geometry, Vol. I., p. 113, enables us to measure with accuracy any part not less than one-sixtieth of the line assumed to repre- As the learner will have to refer to this table when engaged sent 5 degrees. The line required has to bear the same propor- in the construction of a conical projection of any portion of the tion to this line as 42:43 bears to 60, and will be represented by sphere, whether large or small, he should carefully study it, and the dotted line in Fig. 15, drawn midway between the lines endeavour to commit to memory the number of geographical representing 42 and 43, and a little nearer to the former than miles under every fifth parallel of latitude, counting from the to the latter. In Fig. 16 a larger diagonal scale is given, from equator, that is to say, the 5th, 10th, 15th, 20th, etc.






of Lat.






a man



an overseer



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Greek anthology a remarkable mention here of sneezing in an epigram

upon one Proclus."--Brown, “Vulgar Errors." GREEK STEMS (continued).

“The contentions of the Eastern and Western churches about this LANGUAGES have their distinctive peculiarities which fit them subject are but a mere logomachy, or strife about words."— Bishop for some special service in the great workshop of humanity. Bramhall. The numerous broad and open vowels of the Italian makes it

GREEK STEMS. specially suitable as the language of song. The strength and Greek Words. Pronunciation. Meanings. Stems. English Words, dignity of the Latin render it a good organ of civil government. Ανθρωπος an-thro'-pos

anthrop misanthropy. The French, as being light and graceful, is unequalled as a



miso mizogamist. medium of conversation. The swell and pomp of the Spanish

Γαμος gam'-os marriage

bigamy. both represent and symbolise the people by whom it is spoken. Bi (bis) signifies twice, so that bigamy is the state of being Two or three languages possess almost every variety of excel. twice married. lence. Of these, the lowest in the scale is the English, which

No bigami-that is, none that had been twice married, or such as is distinguished alike for power, expressiveness, delicacy, and married widows-were capable of the benefit of clergy, because such music; yet it must, in these high qualities, yield to the German, could not receive orders.”—Burnet, History of the Reformation." which, in its turn, is surpassed by tho Greek, the nearest Bigamy, according to the canonists (the doctors of the ancient approach to perfection to which human language ever attained, ecclesiastical law), consisted in marrying two virgins successively, except, probably, the Sanscrit, or sacred language of the one after another, or once marrying a widow."— Blackstone, " CostBrahmins.

mentaries." As one result of its excellence, the Greek has adapted itself Bigamy, as punished by the English law, is the crime of with equal care and precision to the constantly growing demands having two wives at the same time. of science. On its native soil, and while yet spoken in its purity, the Greek tongue had gained the power of expressing the widest

GREEK STEDS. generalisations, and the nicest distinctions of thought. Its Greek Words. Pronunciation. Meanings. Stems. English words. resources for setting forth the truths of physical science were,

Αρχη ar-che or ar'-ke beginning, chief arch archbishop. in classical times, but very partially put to the test. In the ETICKOTOS e-pis-kop-os

episcop bishop. pages of Cicero, however, we learn how much indebted Rome ApıOuos a-rith'-mos a number arithm arithmetic, was to the Greek for terms of art and of moral and intel- Αστρον as-tron

a star

astro astronomy. lectual disquisition. At the true birth of science, after the Νομος nom'-os

a law

anomaly. revival of letters, the Greek, being cultivated anew, afforded a ATMOS at-mos



atmosphere. most appropriate vehicle for the communication and interchange Σφαιρα sphai'-ra or sphi’-ra a ball or sphere spher spherical. of the new truths which continued to break upon the world

Αντος au'-tos or or'-tos self

auto autograph. in great profusion ; and now, by the creation of several sciences Γραφη graph'-e or graf-fe writing graph calligraphy wholly unknown of old—such as chemistry, botany, physiology,


beautiful kal kaleidoscope. Ειδος eit-dos

a form cido conchology, magnetism, etc.-our scientific vocabulary, with all


Βαπτω its multiplicity, its precision, and conciseness, is found to consist,


I dip

bapt baptism. Βαρος bar'-os

weight baro barometer, for the most part, of elements supplied by the Greek language.

Μετρον met-ron

a measure metr metrical You have an instance in the first word of the ensuing list,

ghee (g hard) the earth

geometry akovW (a-kow'-o), which is the parent of acoustics, or the science

θερμος ther-mos


thermo thermometer. of hearing. The corresponding science of sight has also, in optics,

BeBlucov bib-li-on

a book biblio bibliography. taken a Greek term. Hence you may infer how important is


a description graph graphical. an exact study of these Greek stems. In some sense, indeed,



bio biography. the learning of a science is the learning of the signification of

Χειρ keir or kire the hand chir chirography. its vocabulary, or list of words; assuredly he that is familiar

Χολη kole


cholé choleric. with the elementary roots of the Greek, will, in proceeding to Xpovos kron'-os


chron chronometer. study science, find himself in possession of a most powerful Χρυσος kru'-sos


chryso chrysolite. auxiliary.

Λιθος lith-os

a stone litho

lithography. GREEK STEMS.

Aeka dek'-a




a tree Greek Words. Pronunciation.

dendron rhododendron. Aevdpov den'-dron, Meanings. Stems. English Words. 'Podos


I hear

rhododendron AvOos an'-thos a flower antho anthology.

Δοξη dox'-e

dogma, opinion dox, dog orthodox, dogmaAogos log'-os a word, a dis- logo logomachy.

Ορθος or'-thos

straight, right ortho orthography (tise.

Apoulos course,


a running drom dromedary. scienco

'IXTOS hip’-pos

a horse

hippo hippodrome. Maxn mach'-e a fight

mach naumachy. Graphé (ypaon), in its modern application, means printing as mackNavs

a ship


well as what is strictly writing; it signifies, indeed, a descrip

tion or representation in general, and so may mean a represenThe word logos (toyos) plays a very important part in the tation by strokes of the pen, or a representation by means of world of ancient Greek thought. It is the term by which the the press. Hence you see the application of the term to litho. word of St. John's Gospel is expressed in the original. Logos graphy. denotes either intelligence, the unuttered thought; or speech, Observe now an instance of the use of the Greek. I had, I the uttered thought. From these radical meanings flow the remember, when I was young, some difficulty in ascertaining, numerous applications of the term. In science, the service and when ascertained in remembering, the exact difference which logos renders is very great. In the preceding list, two between the barometer and the thermometer. My little Greek out of the five examples contain the term. Used in a somewhat came to my aid, and showing me that the former was a measure remote sense, indeed, logos, as signifying science, enters into of weight, and the latter a measure of heat, gave me definite the very designation of many of the sciences. Thus we say and clear ideas which I have never forgotten. theology, philology, astrology, demonology, pneumatology, ane- Biblion (BiBAiov) enters into combination with several words mology, ouranology, nosology, phrenology, etc.

With graphe, biblion, forming bibliography, originates a term In naus (vavs) you have a word common to the Teutonic and which signifies the science of books. With the aid of latria the Celtic elements of language, for the naus of the Greeks is (Aarpeia, la-tri'-a, worship) we have bibliolatry, a word sanctioned the navis of the Latins. Meaning ship, it appears not only in by Coleridge, which may be Englished by book-worship, or rontnautical (Latin, nauta, a sailor), but in navigate (Latin, ago, I worship. Bibliomania, or book-madness, is made up of biblion, drive, guide), navigation, etc. The student, by combining naus, a book, and mania (uavia), the Greek for madness. United to a ship, with mache (uaxn, fight), learns that naumachy denotes poleo (Twew), I seli, it forms bibliopolist, a bookseller ; and with a sea-fight.

theca (Onka), the Greek for a repository, it gives rise to the " Anthology signifies properly a collection of flowers, and in par. French bibliothèque, a repository for books-that is, a library. ticular a collection of flowers or gems of poetry. There is in the Let it also be distinctly mentioned that the Greek biblion is the









apo, from.

source whence we get the name of the book of books-namely, the angle in the picture will be at a, touching the base of the the Bible.

picture. Let us state the question as that the angle does not Language is in one view a record of human errors. The fact touch the PP, but is 1 foot within it (Fig. 20). Draw a line is illustrated in the names of some of what are still, by courtesy, from a at the given angle, 45o. From a draw a e, equal to 1 called sciences, such as astrology, phrenology, etc. It is also foot (still using the scale of 4 feet to the inch), and perexemplified in particular words, as, e.g., choleric, coming from pendicular to the PP; draw e c parallel to PP; c will then xoan (kol-e), bile. The term choleric shows that formerly men be the position for placing the angle. It will be seen that the regarded the bile as the source of anger and passion.

line of contact, marked LC, is drawn from a, because a is the " When choler overflows, then dreams are bred

point of contact for the line cb. Thus the pupil will observe, if Of flames, and all the family of red;

no part of the plan touches the PP, one line must be produced, Red dragons and red beasts in sleep we view,

as b c has been done to a, from which the line of contact is For humours are distinguished by their hue."-Dryden. drawn perpendicularly to the PP, and the point of contact, PC, Accordingly, dejection or habitual sadness was termed melani is brought down to the base of the picture, from which the choly, or black-bile.

perspective view of the line b' d is drawn to its v P. The visual

ray from c to s P will determine upon B' pc the position of the EXERCISES IN COMPOSITION.

angle d' in the picture—that is, the perspective distance of one Words with thoir Prepositions to be formed into sentences.

foot within the picture; the rest will be the same as in Fig. 19. Allude to, F. R. lud, play.

The above, and also the greater portion of our previous Alteration in, of, alter, another.

lessons, is a part explanation of one system of the groundAmbition of, ambit, a canvassing,

plan method; we have introduced it first, and said thus Amenable to, amener, to bring.

much upon it, more for the sake of clearing up technicalities Analogous to, Analogy to, between, analog, similar.

than for any other reason. It is a beginning from which we Angry with a person,

intend gradually to lead our pupils into deeper water, and Angry at a thing, angor, choking.

we hope by this course of treatment to make the subject easier Annex to, nexa, a link.

to comprehend. We now intend to take up another line of Animadvert on, animus, mind.

explanation for the same purpose, and here we especially ask vert, to turn,

for the close attention of our pupils whilst we say a few words Antecedent to, ante, before.

upon the way we wish them to proceed. We desire to make ced, to go.

our observations as clear as we can, however difficult it may anti, against. Antipathy to, against,

be to do so; therefore we ask them to accompany us slowly, path, feeling. Anxious about, ans, pain.

and not to feel discouraged if they have to read our instruc

tions more than once. In some of our lessons on drawing, Apologise for, logo, discourse.

we introduced some perspective problems in order to give Appeal to, appel, to hurry to.

the why and wherefore of the reasons for the practice and ad, to.

methods we recommended; it is true that we could have Appertain to, per, through.

simply stated how, and in what direction a line was to be tene, to hold.

drawn, and the pupil might have understood the instruction, Applicable to, plic, to fold, grasp.

and have done as he was directed very satisfactorily; but we Apply to, Apprehension of, prehend, to take hold,

felt it was our duty not to leave him with such superficial Appropriate to, fropr, one's own.

guidance, but open out to him the reasons for these direcApprove of, prob, good.

tions, because if he understood them, he was then furnished Argue with, against, argue, proof.

with a key to innumerable other facts and positions, thus

enabling him to dispense with oft-repeated explanations, varied To improve yourself in simple composition, make a report of in some respects only by the difference there might be in the following anecdote :


This way of proceeding entails a greater amount of difficulty "A friend of mine," says Dr. Bailly, “a man of understanding in the explanation, and necessarily a greater amount of attention and veracity, related to me these two facts, of which he was an eye and study on the part of the pupil; but there is this satisfaction witness. He had an intelligent ape, with which he amused himself attending it, the subject becomes in proportion more interesting, by giving it walnuts, of which the animal was extremely fond. One and a more solid, extensive, and really useful amount of knowday he placed them at such a distance from the ape that the animal, ledge is acquired. These are gains well worth the additional restrained by his chaid, could not reach them. After many useless efforts to indulge himself in his favourite delicacy, the ape happened and pupil, and certainly ought not to be passed over by either.

care and painstaking necessarily incumbent upon both master to see a servant pass by with a napkin under his arm; he immediately seized hold of it, whisked it out beyond his arm to bring the .We are about to follow the same plan again, as far as possible, nuts within his reach, and so he obtained possession of them.His in explaining the reasons for the directions we shall give as we mode of breaking the walnut was a fresh proof of the animal's in- proceed with our lessons. Should difficulties arise, there will ventive

powers; he placed the walnut upon the ground, let a great be no occasion to stop. Let the pupil proceed according to the stone fall upon it, and so got at its contents. One day the ground rules laid down in the problems, and very likely, after he has on which he had placed the walnut was so much softer than usual, done a few, and again returned to those he stumbled over, he will that, instead of breaking the walnut, the ape only drove it into the find they have become clear and simple, and that his future earth, What does the animal do ? He takes up a tile, places the walnut upon it, and then lets the stone fall while the walnut is in difficulties in perspective to beginners is to understand how

course will be at once pleasant and easy. Oise of the great this position." --Sydney Smith.

several planes are brought together upon one plane—that is, the

sheet of paper upon which we draw the picture. We will take GEOMETRICAL PERSPECTIVE.-IV.

la point and place it in its perspective position, and further

illustrate it by an cidograph. First, with regard to the several PROBLEM VII. (Fig. 19).—Two lines, each 5 feet long, form a planes we speak of. The first plane is the plane or surface of right angle ; the angle touches the PP, and is opposite the eye ; the picture (see Fig. 21), PPPP; secondly, the ground-plane, each line is 45° with the PP.

or surface of the ground upon which the object, a, is lying i In this case tvo vanishing points must be found, as there are and, thirdly, there is the plane passing through the eye and two lines in the plan, drawn in different directions. As this is but the picture-plane, its trace being shown by the horizontal line, a repetition of Problem I., Fig. 7, drawn each way, there will or line of sight, parallel with the ground, upon which is drawn be no necessity for our repeating it; only we wish to direct the the semicircle DE' E DEP. The letter E means the eye, or in other attention of the pupil to the position of the eye being opposite words, its position with regard to the object and the picturethe angle. Therefore, the line drawn from the angle to the plane ; De and be mean the distance of the eye from the base of the picture, perpendicularly to the picture-plane, serves, picture - plane thrown round upon the HL, because De" and first, to place the station-point ; secondly, acts as a visual ray; DE' are the same distance from ps (the point of sight) as E is and thirdly, as a line of contact. Consequently, the position of from Ps, being the extremities of a semicircle drawn through

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E from Ps as a contre. Now this horizontal plane through the with the picture-plane. From H and I draw perpendiculars to the eye, E, is turned up upon the perpendicular or picture-plane; picture-plane, and proceed with each extremity as was done with mark the course of the dotted arc from E to E, therefore P8 E the line A B; h i will be the perspective of the given line 1 1. is equal to PS E. Thus, it will be seen, two planes are re- The following remarks upon the line K I will refer to all the duced to one. The result is shown also in Fig. 22. Now we lines similarly drawn—that is, perpendicularly to PP; because the must turn the ground-plane upon the picture-plane. First, let line ki is perpendicular to PP, therefore the perspective repreus repeat a remark or two made in Lesson II., Fig. 5 (page 225). sentation of that line is drawn to the point of sight, viz., K i PS, The line from E to A (Fig. 21) is a visual ray cutting the picture and somewhere upon that line is the position of 1 in i found first plane in B; B then is the picture of the point a. The line from a by drawing from the centre k the arc i L, and joining L with to s P, on the ground-plane, is the ground plan of this visual ray; DE'; this last line cutting K Ps in i, fixes the perspective view therefore a perpendicular line from c (where a SP cuts the base of 1; the same may be repeated for h; ih being joined gives the of the picture) drawn to the line E A, determines B, the picture perspective view of H I. of a, the object. The pupil will now perceive there is a plane PROBLEM IX. (Fig. 24).-Draw the perspective view of a perpendicular to the ground, and also to the picture-plane, pavement composed of square slabs, the edges of which shall measure upon which the distinctive points E, PS, B, C, A, and s P are 1-5 feet; height of eye and distance as before. placed; but first make C A2 equal to c A, as shown in the arc Let A B be the total width of all the slabs, and A1, 1 2, 23, from A to Aạ; this brings the ground-plane upon the picture- 3 4, 4 B, each equal to 1.5 feet. Draw lines from each of these plane (in the same way as we turned E to E 2). Compare divisions to the point of sight : upon the horizontal line set off Fig. 22 with Fig. 21 ; Ao will be seen in both figures. From c, in from Ps to DE 10 feet. From each of the divisions on A B, viz., Fig. 22, draw the arc A’ A'. This will be recognised in Fig. 21 ; A, 1,2, etc., draw lines to DE; where these lines intersect those

Fig. 26

Fig. 25

DE 2


and the line AS to D EI will also be seen in both figures ; there. | drawn from the given divisions a, 1, 2, etc., will be found the fore the line E B A in Fig. 21 is turned round upon the picture. angles from which are drawn the opposite sides of the squares, plane, and represented by A3 B DE', shown also in Fig. 22. viz., 5, 6, 7, 8, 9. We must here observe, as will be seen in Fig. Thus the perspective projection B of the point A on the ground 24, that al lines which retire at an angle of 45°

with the PP have is determined." We have remarked (see Lesson II.) that points the distance-point for their point of sight. For if one side of a are the extremities of lines; and if we can determine the square is parallel with the P P, the other side will be at right positions of points in the picture, we can represent straight angles with the P P; therefore the diagonal of the square will lines by uniting these points. Our pupils will also recollect be 45° with PP. that we have said, “all lines which are perpendicular to the PROBLEM X. (Fig. 25).-Draw the perspective view of a square, picture-plane have the point of sight for their vanishing the sides of which are 3 feet in length, 2 feet from the Pp, and point." Let these observations be borne in mind as we proceed. one side at an angle of 50° with the Pp.

PROBLEM VIII. (Fig. 23).— A straight line, A B, 5 feet long, is Draw a b at the given angle. Find the point c according to perpendicular to the picture-plane lying on the ground, and 1 foot Figs. 13 and 14, Lesson III., and Fig. 20. Construct the square from it; height of eye, 5 feet, and distance from the picture-plane, cdef, and from each angle draw perpendicular lines to the PP, 10 feet. Scale, 4 feet to the inch.

and from thence vanishing lines to the Ps. In these several Draw the line CA B perpendicularly to PP, make c A 1 foot, vanishing lines find the projected angles of the square as in and AB 5 feet.

Draw nl parallel with pp, and 5 feet from Problem VIII., Fig. 23; between these points respectively draw it. Draw PS E perpendicularly to a l, and 10 feet long. straight lines which will produce the perspective representation From Ps, with distance PS e, describe the semicircle DE' E DE of the square. In the next problem we only give the proposition From c, and with the distance ca, draw the arc A D. From and the diagram, trusting the pupil will be able to work it, as C; again draw the arc B F, join F and D with DE', also draw the explanation would be a repetition of Problems VIII. and X.

Between the intersections of C PS with the PROBLEM XI. (Fig. 26).—Draw the perspective view of a parallines from F and

D to be will be the perspective of A B, lelogram 5 feet long, 3 feet broad, one edge at an angle of 40° with viz., a b. Let the line 1 I be 5 feet long, and at an angle of 50o the Pp, and the nearest angle 1 foot within, or from the pp.

the line o PS.

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