338 



KNOWLEDGE. 



September, 1913. 



tan 90° — 



given the differences between consecutive values of 



180 °\ n . (am 180° \ 



). It tan (90° — — - ) were a 



linear function of x, these would all be the same. 



This is evidently not the case, the difference tending 



to become less as x is increased. On the other hand, 



the differences do not vary very much after x := 4 is 



/ 180° \ 

 passed, and if the function y = tan f 90° — ) is 



graphed, the curve obtained does not differ very 

 greatly from a straight line after the point corre- 



Figure 367. 



sponding to * = 4 is passed. (See Graph A, Figure 

 367.) The above method of drawing polygons gives 

 the centres of the square and hexagon accurately; 

 that is to say, this method gives y = 1-000 when 

 x = 4, and y = 1 -732 when x = 6. Since an equal 

 increase in x (the number of sides to the polygon) 

 produces an equal increase in y (the distance of the 

 centre from the mid-point of the side) according to 

 this method, it follows that y =: mx -\- c, where 

 m and c are constant quantities. Consequently 



4 m + c = 1 • 000, and 6 m + c = 1-732, 

 whence c = — -464, and m = -366. 



Therefore y = • 366a; — ■ 464. The values of y 

 thus calculated are given in Column 4 of Table 59, 

 and the differences between them and the true 



values of y (as given by tan ( 90° — J), correct to 



the third decimal place, are shown in Column 5. As 



is evident, these differences are comparatively small 



between x = 3 and x = 9. After this they become 



increasingly large, as can also be seen by comparing 



/ 1 80°\ 

 the graphs of y = tan ( 90° J and y = -366 x 



— -464 (Graph B in Figure 367), so that this 

 method, whilst fairly accurate for polygons with less 

 than 10 sides, is not satisfactory when the number of 

 sides exceeds 9. 



The distance between consecutive centres is 

 assumed in the above method to be -366 x half 

 length of side. Referring to Column 3 of Table 

 2, however, it is obvious that this difference, between 

 x=b and ^=20, is more nearly equal to ^ x half 

 length of side. On this fact we base the three 

 following methods, each of which has special 



advantages. We shall refer to the method discussed 

 above as Method 1. 



Method 2. Let A B (see Figure 364) be one side 

 of the required polygon. Bisect A B in M, and at 

 M draw M P perpendicular to A B, and of indefinite 

 length. With centre B and radius B A describe an 

 arc cutting M P in 6. This point obviously gives the 

 centre of the circle circumscribing the regular 

 hexagon drawn on A B as base (i.e., y = 1 ■ 732 when 

 x=6). Trisect MB (or AM) and, commencing 

 from 6, mark off divisions 7, 8, 9, 10, and so on, 

 along 6 P, each equal to J M B. These points give 

 the centres of the circumscribing circles. The 

 polygons are then drawn as in Method 1. In Figure 

 366 are shown polygons containing 7, 9, 12, 14, and 

 17 sides thus obtained. The first four are very 

 accurate, the last is affected by a slight error. The 

 equation connecting x and y for this method can be 

 obtained thus : Let y=mx + c. Then clearly 



m — -3, and 6w 4- c = 1 • 732 ; 

 whence c = — -268. 



Therefore y=-3x — 268. The values of y thus 



calculated are given in Column 6 of Table 59, and 



the differences between them and the true values of 



y are shown in Column 7. In no case between 



x=4 and #=20 does the difference exceed -085, so 



that this method is a good method for any polygon 



containing 4 to 20 sides. In every case, moreover, 



save between x=3 and x = 5 (which cases are not of 



importance), this difference is less than with Method 1. 



Method 2 is especially accurate when the number of 



sides is 11, 12, 13, 14, or 15. Graph C in Figure 367 



represents the equation corresponding to y = -3x 



— 268. It will be noticed how closely it lies to 



/ 180°\ 



V=tan(90°-— ). 



Table 58. 

 Angles of Regular Polygons. 



