505 



instead of having points like the beam compass, has two micrometer 

 microscopes adjustable to different distances, as aliquot parts of an 

 arc or line to be divided. For the purpose of marking on the surface 

 of the instrument any assumed interval between the microscopes, 

 certain thin pieces of small brass, carrying dots, are employed, with 

 a clamp for fixing them to any part of the limb, and with proper 

 screws for adjusting them to any precise position. 



In addition to these, which are adapted for finding aliquot parts of 

 the circle under division, the apparatus is provided with a cutting 

 point of the usual construction, and a third microscope fixed always 

 in the same position, for the purpose of observing a certain line first 

 laid down as a permanent point of reference, and for watching the 

 performance of the cutting-point, while it marks the successive in- 

 tervals, determined by means of the two adjustable microscopes. 



As a specimen of the method in which this apparatus is to be used, 

 the author describes the series of divisions and subdivisions which he 

 thinks most convenient in a circle of two feet diameter. 



He first divides into five parts of 72 each. 



At the same tune that the fixed microscope is kept steadily over 

 the original line of reference, two adjustable dots are placed, by esti- 

 mation, at the distance of 72 from each other, and in such a position 

 with reference to the cutting-frame as not to interfere with its motion. 

 The two adjustable microscopes having next been placed so that their 

 wires exactly correspond with these dots, the circle is made to re- 

 volve so that the second dot shall come under the first microscope 

 when a third dot is to be placed under the second microscope ; and 

 in the same manner a fourth and a fifth dot in succession, till the 

 fifth interval can be compared with the distance between the micro- 

 scopes, when the first dot should be found to correspond exactly with 

 the second microscope : if it be otherwise, the difference must be 

 measured by the micrometer ; and the distance between the two mi- 

 crometers, originally assumed at 72, must be corrected by one fifth 

 part of the error so found, and must subsequently be ascertained to 

 be correct, by a careful repetition of the same operation of the ad- 

 justable dots. When the fifth interval has thus been made accurately 

 to correspond with that between the adjustable microscopes, the ori- 

 ginal point of reference is then to be returned to its position under 

 the fixed microscope, and the divisions are to be marked in succession 

 as the series of dots are made to appear under the wires of the ad- 

 justable microscopes, which are now known to be accurately 72 

 from each other. 



The next step taken is to divide each of these intervals into three 

 parts of 24 each, and again trisect into parts of 8. These, by re- 

 peated bisection, are reduced to half degrees ; and these again, by 

 trisection, are divided into spaces of 10' each. 



In the performance of the latter steps of this subdivision, the author 

 suggests the periods when it may be found convenient to bisect some 

 larger odd number of divisions, instead of taking a single one for bi- 

 section, on account of the interference of the microscopes with each 



