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104 LECTURE X. 



vided,for example, in the ratio of 2 to 3, 3 to 4, or 4 to 5, or of 10 to 1 1, !S 

 oris, at pleasure. (Plate VI. Fig. 90.) 



The use of the sector depends also on the properties of similar triangles. The 

 scale of equal parts, which is laid down on each leg, beginning from the cen- 

 tre, serves to determine the length of the legs of two equilateral triangles, 

 in any required proportion to each other, according to the division which we 

 mark, and the transverse distances from the corresponding points are neces- 

 sarily in the same proportion. Thus, if we have any line in a figure which we 

 wish to call three feet, or three inches, we may take the interval with a pair 

 of common compasses, and open the sector to such an angle, that it may ex- 

 tend from the third division of one leg to that of the other ; then all the other 

 divisions of the scale will furnish us with the lengths corresponding to any 

 distances that we may wish to lay down. The other scales usually engraved 

 on the sector are principally intended for trigonometrical calculations on 

 similar principles. (Plate VII. Fig. 91.) 



The mag-nitude of angles admits an easy determination and description, by 

 the comparison of the respective arcs with a circle, or with a right angle. 

 We may divide an angle geometrically, by continual bisection, into parts as 

 small as may be required, and by numbering these parts, we may define any 

 angle, with an error smaller than any assignable quantity. Bisections of 

 this kind are sometimes actually employed in the construction of instruments; 

 for instance, in one of the arcs of the mural quadrant of the observatory at 

 Greenwich, the right angle is divided into 96 parts, by the continual bisec- 

 tion of one sixth of the circle. There arc also some practical methods of di- 

 viding angles into three or more equal parts, which are sufliciently accurate 

 for many purposes, although it is well known that in theory the perfect tri- 

 section of an angle is beyond the reach of plain geometry. This trisection is, 

 necessary in the common division of tlie circle into 360 degrees, a number 

 which was probably chosen because it admits a great variety of divisors, and 

 because it nearly represents the diurnal and annual motion of the sun among 

 the stars. The circle being^ divided into 6 parts, the chord of each of which 

 is equal to the radius, these parts are divided into 60 degrees, each degree 

 into 60 minutes, and each minute into 60 seconds : further than this we can- 



