that the space thus assumed is not taken too long in proportion to the base line of the map, and set off from E along the straight line E B eight of these spaces, and number the points, beginning with E, 35, 40, 45, 50, 55, 60, 65, 70, 75, as in the figure, and through the point marked 75 draw the straight line G H parallel to C D. The line G H will be the limit of the map towards the top, and the points numbered upwards in succession from E will be the points through which will pass the parallels of latitude corresponding with the numbers. To find the centre from which to draw the parallels of latitude, measure four spaces upwards from the point marked 75, along A B, and number them 80, 85, 90, and F. The point F, just five degrees higher than the pole marked 90, is the centre from which the parallels of latitude are to be described. This point corresponds to the point м in Fig. 14, in which the sphere is supposed to be entered by a cone passing through it in parallels 45 and 65, the spaces from 45 to M, along the line L M, in Fig. 14, and the spaces from 45 or P to F, along the line A B, in Fig. 17, corresponding in number, as the reader may ascertain on comparing them. The actual distance of the point F beyond the pole is 4° 30′ 30′′, when determined by calculations involving a knowledge of trigonometry; but for maps on a small scale it is near enough to consider it as being equal to five degrees, and therefore to the space assumed to represent five degrees in the construction of the projection. The point F being thus determined, the parallels of latitude may be described through the points marked on the central meridian with a finely pointed pencil, or, as there is no absolute occasion to describe these arcs until the limits of the map on either side are determined, it will be sufficient to draw a single arc o P Q through the point marked 45 or P. This arc should be drawn for some distance beyond the ultimate limits of the map on either side. We now proceed to draw the meridians, and to determine their position. The learner will have to refer to the table at the end of this lesson, which shows the number of geographical miles contained in a degree of longitude under each parallel of latitude, supposing the earth to be a perfect sphere in form.

It will be remembered that the cone on which our map of Europe is projected was supposed to pass through the sphere in the parallels of 45° and 65°, and that the measurements on the cone along these lines are exactly equal to measurements along the same lines on the sphere, or, in other words, that the degrees of longitude on these parallels, both on the cone and the sphere, are exactly equal. On looking at the table, we find that a degree of longitude under the parallel of 45° is equal to 42:43 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 proportion to the line which we assumed at first to represent 5. 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 either side of the central meridian, through which the other meridians may be drawn from the point F. It would do equally well to find a line bearing the same proportion to the line assumed to represent 5 degrees as 25:36 bears to 60, and to set off spaces equal to this line on either side of the central meridian along the arc representing the parallel of 65°; but it is always safer, and ensures a higher degree of accuracy, to deal with the larger arcs and spaces instead of the smaller.

To enable us to find lines bearing the required proportions to the line originally assumed to represent 5 degrees, we must take a straight line exactly equal to it, as in Fig. 15, and on it construct a square. The sides of this square must be divided into six equal parts, and numbered upwards at the points of section from 0 to 60, while the top and bottom must be divided into ten equal parts, the points of section between the extremities being numbered from 1 to 9. Lines must then be drawn diagonally across the square, from 0 on the left hand to 10 on the right hand, etc., and perpendicular lines parallel to the sides through the points of section numbered 1, 2, 3, etc. This diagonal scale, constructed on the same principle as the scale shown in Lessons in Geometry, Vol. I., p. 113, enables us to measure with accuracy any part not less than one-sixtieth of the line assumed to represent 5 degrees. The line required has to bear the same proportion to this line as 42-43 bears to 60, and will be represented by the dotted line in Fig. 15, drawn midway between the lines representing 42 and 43, and a little nearer to the former than to the latter. In Fig. 16 a larger diagonal scale is given, from

which the reader may construct a projection for a map of Europe, taking the dotted line A B to represent the distances to be set off along the arc representing the parallel of 45°; but it will be better for him to construct scales for himself, much larger in size than the largest which we have given in Fig. 16. Distances equal to the line a b that represents 42:43 on the small scale in Fig. 15, must now be set off on either side of the central meridian, represented by the straight line A B, along the arc O P Q, and straight lines must be drawn from F with a fine drawing-pen through the points thus obtained. The dotted lines between F and the top of the map need not be drawn by the learner. The remaining arcs must then be drawn with a compass pen, and the limits of the map to the east and west determined by drawing the straight lines K L, MN at right angles to the base line CD, the former a little to the left of the meridian 5° west longitude, and the latter a little to the right of the meridian 45° east longitude. The border lines should then be drawn as shown in the engraving. The double lines at the sides and top and bottom of the inner space, which contains the map, should be divided into single degrees and ruled, as in the figure, to present a distinction of colour, and thus afford a ready means of counting and measuring degree lines not marked and numbered on the map. The meridians should be numbered in the border at the top and bottom, and the parallels of latitude at the sides. The Arctic Circle must be inserted in the form of a dotted line at the distance of 1° 30′ above the parallel of 65°. A blank space should be left in the upper left-hand corner, or the lower right-hand corner, for the title and scale of geographical and British miles. To construct these scales, it must be remembered that 60 geographical miles are equal to 69.07 British miles, or that the line which was at first assumed as being equal to 5 degrees, represents 60 × 5, or 300 geographical miles, and 69.07 × 5, or 345 British miles, very nearly.

In order to fix the position of places with accuracy, the student is advised to divide the field of his map by pencil lines into spaces of a degree each way, as shown in the lower part of Fig. 17. This, however, can only be done when the map is on a sufficiently large scale. Learners are cautioned to use Indian ink instead of common ink in drawing maps, as the ordinary ink will run and spoil the map where a final wash of colour is given to the sea, and the boundary lines are distinguished by contrasting tints. TABLE SHOWING THE NUMBER OF GEOGRAPHICAL MILES IN

A DEGREE OF LONGITUDE UNDER EACH PARALLEL OF LATITUDE, THE EARTH BEING SUPPOSED TO BE A PERFECT SPHERE.

0 60.00 69.07

1 59.99 69.06

2 59-96 69.03

3 59.92 68.97 4 59.85 68.90

5

59 77

6

59 67

68.81 68 62

7 59:55 68:48

23 55 23 63.51 24 54-81 63:03 25 54-38 62:53 26 53-93 62.02 27 53:46 61.48 28 52-98 60.93 29 52 48 60:35 30 51-96 59.75 31 51:43 59*13 32 50-88 58.51 33 50-32 57.87 31 49-74 57-20 35 49 15 56:51 36 48-54 55-81 37 47.92 55:10 38 47.28 54:37 39 46 63 53:62 17 57-38 65-98 40 45 96 52.85 18 57:06 65.62 41 45 28 52:07 19 56 73 65.24 42 44 59 51-27 20 56:38 64.84 43 88 50 46 21 56:01 64:42 44 43 16 49-63 22 55 63 63.97 45 42 43 48.78

11

12

15

57 96

16

57 68

8 59:42 68:31 9 59 26 68:15 10 59:09 67.95 58 90 67-73 58 69 67:48 13 58:46 67.21 14 58 22 66.95 66.65 66-31

43

As the learner will have to refer to this table when engaged in the construction of a conical projection of any portion of the sphere, whether large or small, he should carefully study it, and endeavour to commit to memory the number of geographical miles under every fifth parallel of latitude, counting from the equator, that is to say, the 5th, 10th, 15th, 20th, etc.

LESSONS IN ENGLISH.-XXV.

GREEK STEMS (continued).

LANGUAGES have their distinctive peculiarities which fit them for some special service in the great workshop of humanity. The numerous broad and open vowels of the Italian makes it specially suitable as the language of song. The strength and dignity of the Latin render it a good organ of civil government. The French, as being light and graceful, is unequalled as a medium of conversation. The swell and pomp of the Spanish both represent and symbolise the people by whom it is spoken. Two or three languages possess almost every variety of excellence. Of these, the lowest in the scale is the English, which is distinguished alike for power, expressiveness, delicacy, and music; yet it must, in these high qualities, yield to the German, which, in its turn, is surpassed by the Greek, the nearest approach to perfection to which human language ever attained, except, probably, the Sanscrit, or sacred language of the Brahmins.

As one result of its excellence, the Greek has adapted itself with equal care and precision to the constantly growing demands of science. On its native soil, and while yet spoken in its purity, the Greek tongue had gained the power of expressing the widest generalisations, and the nicest distinctions of thought. Its resources for setting forth the truths of physical science were, in classical times, but very partially put to the test. In the pages of Cicero, however, we learn how much indebted Rome was to the Greek for terms of art and of moral and intellectual disquisition. At the true birth of science, after the revival of letters, the Greek, being cultivated anew, afforded a most appropriate vehicle for the communication and interchange of the new truths which continued to break upon the world in great profusion; and now, by the creation of several sciences wholly unknown of old-such as chemistry, botany, physiology, conchology, magnetism, etc.-our scientific vocabulary, with all its multiplicity, its precision, and conciseness, is found to consist, for the most part, of elements supplied by the Greek language. You have an instance in the first word of the ensuing list, akova (a-kow-o), which is the parent of acoustics, or the science of hearing. The corresponding science of sight has also, in optics, taken a Greek term. Hence you may infer how important is an exact study of these Greek stems. In some sense, indeed, the learning of a science is the learning of the signification of its vocabulary, or list of words; assuredly he that is familiar with the elementary roots of the Greek, will, in proceeding to study science, find himself in possession of a most powerful auxiliary.

Bi (bis) signifies twice, so that bigamy is the state of being twice married.

"No bigami-that is, none that had been twice married, or such as married widows-were capable of the benefit of clergy, because such could not receive orders."-Burnet, "History of the Reformation." "Bigamy, according to the canonists (the doctors of the ancient ecclesiastical law), consisted in marrying two virgins successively, one after another, or once marrying a widow."-Blackstone, mentaries."

Com

chronometer.

chryso chrysolite.

lithography.

deca decalogue.

dendron rhododendron.

rhodo

dogma, opinion doz, dog orthodox, dogma

Navs

The word logos (Aoyos) plays a very important part in the world of ancient Greek thought. It is the term by which the word of St. John's Gospel is expressed in the original. Logos denotes either intelligence, the unuttered thought; or speech, the uttered thought. From these radical meanings flow the numerous applications of the term. In science, the service which logos renders is very great. In the preceding list, two out of the five examples contain the term. Used in a somewhat remote sense, indeed, logos, as signifying science, enters into the very designation of many of the sciences. Thus we say theology, philology, astrology, demonology, pneumatology, anemology, ouranology, nosology, phrenology, etc.

In naus (vavs) you have a word common to the Teutonic and the Celtic elements of language, for the naus of the Greeks is the navis of the Latins. Meaning ship, it appears not only in nautical (Latin, nauta, a sailor), but in navigate (Latin, ago, I drive, guide), navigation, etc. The student, by combining naus, a ship, with mache (uaxn, fight), learns that naumachy denotes a sea-fight.

"Anthology signifies properly a collection of flowers, and in par. ticular a collection of flowers or gems of poetry. There is in the

dromedary.

hippodrome.

Graphe (ypapn), in its modern application, means printing as well as what is strictly writing; it signifies, indeed, a description or representation in general, and so may mean a represen tation by strokes of the pen, or a representation by means of the press. Hence you see the application of the term to litho graphy.

Observe now an instance of the use of the Greek. I had, I remember, when I was young, some difficulty in ascertaining, and when ascertained in remembering, the exact difference between the barometer and the thermometer. My little Greek came to my aid, and showing me that the former was a measure of weight, and the latter a measure of heat, gave me definite and clear ideas which I have never forgotten.

Biblion (BiBXiov) enters into combination with several words. With graphe, biblion, forming bibliography, originates a term which signifies the science of books. With the aid of latria (Aarpeia, la-tri'-a, worship) we have bibliolatry, a word sanctioned by Coleridge, which may be Englished by book-worship, or u worship. Bibliomania, or book-madness, is made up of biblion, a book, and mania (uavia), the Greek for madness. United to poleo (wλew), I sell, it forms bibliopolist, a bookseller; and with theca (Onka), the Greek for a repository, it gives rise to the French bibliothèque, a repository for books—that is, a library. Let it also be distinctly mentioned that the Greek biblion is the

E from PS as a centre. Now this horizontal plane through the eye, E, is turned up upon the perpendicular or picture-plane; mark the course of the dotted arc from E to E2, therefore PS E2 is equal to PS E. Thus, it will be seen, two planes are reduced to one. The result is shown also in Fig. 22. Now we must turn the ground-plane upon the picture-plane. First, let us repeat a remark or two made in Lesson II., Fig. 5 (page 225). The line from E to A (Fig. 21) is a visual ray cutting the pictureplane in B; B then is the picture of the point A. The line from A to s P, on the ground-plane, is the ground plan of this visual ray; therefore a perpendicular line from c (where A SP cuts the base of the picture) drawn to the line E A, determines B, the picture of A, the object. The pupil will now perceive there is a plane perpendicular to the ground, and also to the picture-plane, upon which the distinctive points E, P S, B, C, A, and SP are placed; but first make CA2 equal to c A, as shown in the arc from A to A2; this brings the ground-plane upon the pictureplane (in the same way as we turned E to E 2). Compare Fig. 22 with Fig. 21; A2 will be seen in both figures. From c, in Fig. 22, draw the arc A2 A3. This will be recognised in Fig. 21;

with the picture-plane. From H and I draw perpendiculars to the picture-plane, and proceed with each extremity as was done with the line A B; hi will be the perspective of the given line HI. The following remarks upon the line K I will refer to all the lines similarly drawn-that is, perpendicularly to PP; because the line KI is perpendicular to PP, therefore the perspective representation of that line is drawn to the point of sight, viz., K i PS, and somewhere upon that line is the position of 1 in i found first by drawing from the centre K the arc I L, and joining L with DE'; this last line cutting K PS in i, fixes the perspective view of I; the same may be repeated for h; ih being joined gives the perspective view of H I.

PROBLEM IX. (Fig. 24).-Draw the perspective view of a pavement composed of square slabs, the edges of which shall measure 1.5 feet; height of eye and distance as before.

Let A B be the total width of all the slabs, and A1, 12, 23, 34, 4 B, each equal to 1.5 feet. Draw lines from each of these divisions to the point of sight: upon the horizontal line set off from PS to DE 10 feet. From each of the divisions on A B, viz., A, 1, 2, etc., draw lines to DE; where these lines intersect those

and the line A to D E will also be seen in both figures; there fore the line E B A in Fig. 21 is turned round upon the pictureplane, and represented by A3 B DE', shown also in Fig. 22. Thus the perspective projection B of the point A on the ground is determined. We have remarked (see Lesson II.) that points are the extremities of lines; and if we can determine the positions of points in the picture, we can represent straight lines by uniting these points. Our pupils will also recollect that we have said, "all lines which are perpendicular to the picture-plane have the point of sight for their vanishingpoint." Let these observations be borne in mind as we proceed. PROBLEM VIII. (Fig. 23).—A straight line, A B, 5 feet long, is perpendicular to the picture-plane lying on the ground, and 1 foot from it; height of eye, 5 feet, and distance from the picture-plane, 10 feet. Scale, 4 feet to the inch.

and AB 5 feet.

it.

Draw the line CA B perpendicularly to PP, make C A 1 foot, Draw HL parallel with PP, and 5 feet from Draw PS E perpendicularly to H L, and 10 feet long. From PS, with distance PS E, describe the semicircle DE1 E DE'. From c, and with the distance CA, draw the arc A D. From C, again draw the arc B F, join F and D with DE2, also draw the line C PS. Between the intersections of c PS with the

lines from F and D to DE will be the perspective of AB, viz., a b. Let the line H I be 5 feet long, and at an angle of 50°

drawn from the given divisions A, 1, 2, etc., will be found the angles from which are drawn the opposite sides of the squares, viz., 5, 6, 7, 8, 9. We must here observe, as will be seen in Fig. 24, that all lines which retire at an angle of 45° with the PP have the distance-point for their point of sight. For if one side of a square is parallel with the P P, the other side will be at right angles with the PP; therefore the diagonal of the square will be 45° with PP.

PROBLEM X. (Fig. 25).—Draw the perspective view of a square, the sides of which are 3 feet in length, 2 feet from the PP, and one side at an angle of 50° with the PP.

Draw a b at the given angle. Find the point c according to Figs. 13 and 14, Lesson III., and Fig. 20. Construct the square c def, and from each angle draw perpendicular lines to the PP, and from thence vanishing lines to the PS. In these several vanishing lines find the projected angles of the square as in Problem VIII., Fig. 23; between these points respectively draw straight lines which will produce the perspective representation of the square. In the next problem we only give the proposition and the diagram, trusting the pupil will be able to work it, as the explanation would be a repetition of Problems VIII. and X.

PROBLEM XI. (Fig. 26).—Draw the perspective view of a parallelogram 5 feet long, 3 feet broad, one edge at an angle of 40° with the PP, and the nearest angle 1 foot within, or from the PP.