October 1, 1887.] 



KNOV/LEDGE ♦ 



281 



I proceed to give a brief description of the geometrical 

 processes involved in tlie construction of the figures of 

 plates i. to xiii. of my " Seasons Pictured," conceiving that 

 these processes may have an interest for the mathematical 

 reader. 



Describe a circle, a.¥a' (fig. 3), to represent the outline of 

 the disc, and take the angle aOP, equal to the obliquity of 

 the ecliptic (or nearly 23° 30). Draw Po at right angles to 

 aO, and describe the arc PL with centre o and radius oP. 

 On this arc take the angle PoL, equal to the angle described 

 by the earth around the sun from the vernal equinox, at the 

 moment considered. (For instance, for the live sun-views 

 one month after the equinox in my " Seasons Pictured " the 

 angle would be 30° ; for those one month before the 

 Midsummer solstice the angle would be 60° ; and so on.) 

 Then LP, drawn at right angles to oP, gives 1' the pole of 

 the earth. Thus I'Op' is the polar axis, and W)E' at right 

 angles to POP' is the m;ijor axis of the ellipse which will 

 represent the equator. The half- minor axis Oe is equal to 

 LP* 



To describe an ellipse, having given axes, is a very simple 



points ci and €>, on the ellipse representing the equator, 

 and also on the longitude-circles we require. 



To determine the ellipse corresponding to another parallel 

 originally appearing .as the line cnc', a very similar method 

 is available. The line nn parallel to Po gives the point ii 

 on OP, which is the centre of the ellipse we require ; (!»C', 

 equal in length to cc', and at right angles to (and bisected 

 by) PP', is the major axis ; and since it is clear that this 

 parallel, being (as its name implies) parallel to the eciuator, 

 must be opened to exactly the same proportinnal extent, we get 

 the half-minor axis hk by drawing the line CV- parallel to Et. 

 Describe circles of which CG and Ik are quadrants, and divide 

 the quadrant C(J into three equal parts in 1, 2, as in the 

 former case ; then, the lines nil, n 2 2, being drawn, the 

 lines through 11 and 22 parallel to the axes give the points 

 K,, Ko, on the ellipse and also on the longitude-circles we 

 require. The quarter-ellip.ses Ck, Ee, and the arcs K,f,yi, 

 K„e.ryi, of the two ellipses representing the longitude-circles 

 we require, may now be drawn in : and by carrying on this 

 process for other latitude-parallels and longitude circles we 

 get the complete sets of lines shown in the figures of my 



Fia. :i 



matter. Since it is necessar}', for our purposes, that we 

 should have lonr/itvde-chxles as well as latilude-parallels, 

 the following metho<l, which gives points of both curves at 

 once — that is to say, gives the points where the longitude- 

 circles and latitude-parallels we wish to draw, intersect — 

 is the most convenient : — 



A circle, of which cM is a quadrant, is de.scribed round 

 as centre with radius equal to PL. The quadrant Eji;' is 

 divided into three equal parts in e, and Cj ; and lines are 

 drawn from O to these points, meeting the quadrant «M in 

 /, and/j. Then lines «|«|, e.^e.,, parallel to PP', and lines 

 /,£,, /o€2, parallel to EE', give by their intersection the 



* This is easily shown ; for the globe may be supposed to have 

 assumed its present position (with PO'/" for polar axis) by having 

 been rotated about EOE' in such a manner that P moved along the 

 arc pP (the foreshortened view of an arc equal to ^D' or pH obtained 

 by drawing DPD' at right angles to OP). By this motion the 

 equator, originally seen as a line EOE', would open out into the 

 ellipse EeE', and, since the arc Oe would necessarily be the same as 

 the arc ^^D, tlie line Of %vould clearly be equal to PD. But PD is 

 equal to PL, since (by Euc. III., .S) the square of either of these 

 lines is equal to the rectangle pP.PP. 



Fio. t. 



" Seasons Pictured." The quarter-ellipses cE' and kC, the 

 portion of .an ellipse Ik I' beyond C and C, and K^yj, K.y,, 

 parts of longitude-circles crossing these, are added to show 

 the relation of that part of the construction which has been 

 gone through to the complete figure. It will be gathered 

 that the construction of the figures of my " Seasons Pic- 

 tured " has involved some degree of care and labour — a 

 point which I only mention because tiiose figures have been 

 spoken of as if they had been drawn freehand. 



Of course, the careful construction of the ellipse CkC'k' 

 would give correctly the points / and 7' in which this 

 ellipse meets the cii-cle aEa'E' — points corresponding to the 

 place of sunrise and sunset for places on the latitude- 

 parallel CC. But it is well to determine these important 

 points by a construction founded on the following simple 

 considerations. Since it is clear from what has been said 

 about straight lines passing through aOa' that tho plane of 

 the latitude-parallel cc' (fig. 1) always meets Oa in the same 

 point K (fig. 3), and since the line of intersection of this 

 plane with the plfim aEa'E' must be parallel to the line 

 EOE' (which is the intersection of the plane of the equator 

 with the plane aEa'E'), the line /K/' parallel to EOE' is 



