162 



THE POPULAR EDUCATOR 



object to the eye. Fig. 1 will assist to explain this. Let abed 

 be a slab or board lying upon the ground, let s p be the station 

 point where we are supposed to stand, and E the eye ; / g i h is 

 the plane of glass, or picture plane, through which we see the 

 slab. Lines from each angle of the object passing through the 

 picture plane towards the eye are termed visual rays, and where 

 they pass through the picture plane determine upon the plane 

 the points of the original object ; these points united by straight 

 lines produce the perspective representation required. It must 

 be understood that these visual rays are not limited to proceed 

 from the angles only ; they come from every part at the same 

 time ; but when representing the object we use only those lines 

 which proceed from angles and terminations of lines, and thus 

 determine the proportion of the object in sight upon the picture 

 plane. It will be seen when we come to work the problems that 

 these visual rays are drawn from the various angles and charac- 

 teristic points of the plan of the object towards the station point, 

 s P. It will be noticed, also, that it is not necessary to draw 

 these lines beyond the picture plane, p p, but invariably in the 

 direction of the station point. 



9. The Point of Contact is that point found by continuing in 

 the same direction any original line of the ground plan, until it 

 meets the picture plane ; if the original line touches the picture 

 plane, it then produces its own point of contact. 



10. Line of Contact is a term given to that line which is 

 drawn perpendicularly from the point of contact ; it is some- 

 times called a measuring line, from its being used to point, or 

 mark off, all heights in working a perspective drawing. Of 

 course in this oase the same scale is used as that for laying out 

 the ground plan. 



11. All retiring lines have vanishing points. 



12. All horizontal retiring lines have their vanishing points 

 upon the line of sight. 



13. Allparallel retiring lines have the same vanishing point. 



14. All horizontal lines which are parallel with the picture 

 plane are drawn parallel with each other and with the line of 

 sight. 



15. All horizontal retiring lines forming right angles with the 

 picture plane have the point of sight for their vanishing point. 



16. All lines inclined with the horizon and with the picture 

 plane have their vanishing points above or below the horizontal 

 line, or line of sight, according to the angle they form with the 

 horizon, their vanishing points being always on a line perpen- 

 dicular to the vanishing point upon the line of sight to which 

 they would have retired had they been horizontal. Observe, all 

 heights are set off on the lines of contact ; all horizontal lengths 

 and breadths are arranged on the ground plan. 



As it is our intention to apply these lessons practically that 

 is, to make the drawings according to some given scale it will be 

 necessary to step aside a little from our course, and explain what 

 is meant by a scale, and the method of constructing it, so that 

 any one who wishes to make a perspective drawing of a building 

 or any other object, according to some stated dimensions, may 

 have no difficulty in this respect in carrying it out. A scale 

 is a means by which a proportional measurement of an object is 

 represented ; or, by having a plan of that object, it is a means 

 by which we may obtain an exact idea of all its parts in pro- 

 portion to one another and to the original object. For instance, 

 suppose a room to be 20 feet long and 15 broad, represented by 

 a plan in the proportion of 1 inch to a foot, the drawing or plan 

 will be then 20 inches long and 15 broad ; and if we require 

 single inches in the scale for the plan, the first inch of the scale 

 must be divided into 12 parts. The scale being thus completed, 

 we can measure spaces not limited to feet. Suppose the distance 

 from one corner of the room to the side of a window should 

 measure 6 feet 8 inches, the scale divided as above in the first 

 division will enable us to show that distance on the plan. 



To construct a scale of half an inch to a foot, draw a line of 

 any length, and upon it mark off any required number of half- 

 inches. (See Fig. 2.) Divide the first division into twelve 

 parts, to represent inches, or into four parts to represent spaces 

 of 3 inches, and number the divisions as shown in the figure. 

 To measure 9 feet 9 inches, we must place one leg of the com- 

 passes on nine of the main divisions, and the other on nine of the 

 minor divisions, marked in Fig. 2 from a to b. 



Sometimes scales are much smaller than the above, when the 

 subject is extensive and the drawing small. It is advisable to 

 make our scales generally about 6 inches long they may be 



either a little more or less ; the length is not important, so that 

 there be a sufficient number of parts on the scale to make it 

 useful, but 'the average length of about 6 inches is the most con- 

 venient for general purposes. 



To obtain the average length, we raise or lower both terms, as 

 the case may require, by multiplying or dividing each by the 

 same figure, so that the proportion remains the same : for 

 example, 1 inch to 7 feet, 1 X 6 = 6, 7 X 6 = 42. Therefore 1 to 7 

 is the same proportion as 6 to 42. Again, 14 inches to 100 feet ; 

 this must be lowered, because a scale 14 inches long would be of 

 unnecessary length, therefore 14 -=- 2 = 7, 100 -f- 2 50 ; so that 

 we can make a more manageable scale of 7 inches long to repre- 

 sent 50 feet, which will be the same as 14 to 100. It will be 

 rendered clearer if we propose to make a scale of 1 inch to 88 

 feet. The pupil will see the difficulty of dividing an inch into 

 38 parts and then constructing a lengthened scale from it. To 

 avoid this, we first raise the terms by multiplying both by 6, 

 which will be 6 to 228, and then state the question in the form 

 of a Rule of Three sum. But as we do not wish to go through 

 the trouble of dividing 6 inches into 228 parts, we must find the 

 length of line necessary to include the nearest whole number to 

 228, which is 200, and say as 38 : 1 : : 200 : 5'26. 



It will be seen by this that 5'26 inches to 200 feet is the 

 same proportion as 1 inch to 38 feet, and this simplifies the work 

 in making the scale. To do this we draw a line 5'26 inches long (to 

 measure this distance, see Lessons in Geometry, page 113, Vol. I.), 

 and divide it into two equal parts to represent hundreds, and the 

 first division into ten equal parts to represent tens. (See Fig. 3.) 

 The distance, 170 feet, measured from this scale will be from 

 a to b. To divide a line into any given number of equal parts, 

 see Lessons in Geometry, Problem XII., page 192, Vol. I. 



We will give two other examples, and leave the pupil to 

 practise this method of constructing scales of any given pro- 

 portion. Construct a scale of 1 inch to 13 feet. 13 X 6 = 78. 

 In this case 80 is the nearest whole number to 78, to be stated 

 thus : as 13 : 1 : -. 80 : 6'15 ; therefore draw a line 6'15 inches 

 long, and divide it into eight equal parts to represent tens, and 

 the first division into ten equal parts to represent units. Suppose 

 it were 2 inches to 13 feet, then we should have to raise the 

 number 13 by 3. 13 X 3 = 39 ; 40 would be the nearest whole 

 number in this case ; then as 13 : 2 : : 40 : 6'15. Therefore a 

 line 6'15 inches long is to be divided into four parts to repre- 

 sent tens, and the first division into ten parts to represent 

 units. 



We will now explain how a' b' c' d', on the picture plane / g i h 

 of Fig. 1, is the perspective representation of the square abed, 

 the plan of the square. E is the eye of the spectator when he is 

 standing at s p, the station point ; p S is the point of sight, and 

 H L the horizontal line ; the lines from a b c d to E are the 

 visual rays ; the lines from ab c d to S P are the plans of the 

 visual rays ; from the points where these last lines (the plans of 

 the rays) cut the base of the picture plane, h i, draw perpendi- 

 cular lines to cut the corresponding visual rays in a' b' c' d', join 

 these points respectively, and then will be produced on the 

 picture plane, f g i h, the perspective representation required. 

 This figure is intended only to show how the plan, the eye, and 

 the picture plane are supposed to be arranged with regard to 

 each other, and that the point of sight, P s, is opposite the eye 

 and on the horizontal line, H L, which is on a level with the eye. 



LESSONS IN GREEK. VI. 



THE SECOND DECLENSION. 



THERE are in the Greek second declension two terminations, 

 that in os corresponding with the Latin us, and that in ov corre- 

 sponding with the Latin um. Of the nouns which terminate in 

 os the greater number are of the masculine gender, some are 

 also feminine ; nouns in ov are of the neuter gender, except di- 

 minutive female names, as ^ rAu/cepioj/, Glycerium. 

 The following table presents 



THE CASE-ENDINGS OF THE SECOND DECLENSION. 



Singular. Plural. Dual. 



Nom. ox or 01 a (a 



Gen. ou oiv oiv 



Dat. if ois oiv 



Ace. ov ovs a '>> 



Voc. (os) QV 01 a ca 



