80 LECTURE X. 



The scale of equal parts, which is laid down on each leg, beginning from 

 the centre, 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 corre- 

 sponding points are necessarily 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 extend 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 magnitude 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 construc- 

 tion of instruments ; for instance, in one of the arcs of the mural quad- 

 rant of the observatory at Greenwich, the right angle is divided into 96 

 parts, by the continual bisection of one sixth of the circle. There are 

 also some practical methods of dividing angles into three or more equal 

 parts, which are sufficiently accurate for many purposes, . although it is 

 well known that in theory the perfect trisection of an angle is beyond the 

 reach of plane geometry. This trisection is necessary in the common 

 division of the 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 cannot 

 easily carry the accuracy of our determination, although in calculations 

 we sometimes descend as far as tenths or even hundredths of a second. 

 The decimal division of a right angle, which has been lately adopted in 

 France, appears to have very little advantage for the purposes of calcula- 

 tion, beyond the common method, and its execution in practice must be 

 much more difficult. 



Whole circles, or theodolites, divided into degrees and their parts, quad- 

 rants and sextants, are usually employed in measuring angles ; and protrac- 

 tors, semicircles, and lines of chords, in laying them off. The most convenient 

 of quadrants for general use is Hadley's reflecting instrument,* which is in 

 fact an octant or a sextant, but in which, for reasons depending on optical 

 principles, each degree of the arc is reckoned for two. 



For the graduation of all instruments of this kind, of moderate dimen- 

 sions, Mr. Ramsden's dividing engine is of great utility ;t the instrument 



. * Ph. Tr. 1731, p. 147. f Description, 4to, 1787. Rozier, i. 147. 



