301- Oit a DcXc Sc.iiu/i/ rarii/ljj inve)iU'J 



division of the limb. This, however, supposes that the prisms 

 be made with the utmost precision, and that the axis of the 

 telescope shall be always perpendicular to the common section 

 of the reflecting sides of the prisms. 



With respect to the telescope, it is easy to show, that if its 

 axis be inclined towards the common section of the reflectinp- 

 planes of the prisms, the fourth part of the true angle will 

 have for its sine, the sine of the fourth part of the angle 

 given by the instrument, multiplied by the co-sine of the in- 

 clination of the axis. 



In efi'ect, let (fig. 4) S R, S T, be the two reflecting planes 

 intersected by SQ; let us sujipose a plane SV, which divides 

 this angle, that we draw the right line A B perpendicular 

 to S Q, and the oblique line A D = AB=1. I'rom D draw 

 the perpendicular DH upon B A, and from the points D and 

 B let fall die two perpendiculars DE, BC upon the plane 

 SB; lastly, draw the right lines EA, CA, and HF parallel 

 to C B. Now it is known by the principles of optics, that if 

 B A represent the axis of the telescope, th6 angle formed by 

 two coincident objects by reflection is quadruple the angle 

 CAB, or twice the motion of the index. But if the angle 

 has the obliquity DA, the true angle is the quadruple of 

 the angle D A E, although the index gives the same angle as 

 it had marked before. 



To show the error, then, it is sufficient to determine the value 

 of the angle DAE, by means of the known angles CAB and 

 B AD ; by the construction we have A H : H E : : A B : B C. 

 or because H F= D E cos D A H : sin D A E : : 1 : sin C AB; 

 from whence we shall, as I have said, have sin D A E = 

 binCABcosDAH. 



By this formula we perceive that the greatest error must 

 take place when CAB = 45°. In this case, if the axis of the tele- 

 scope has only one degree of inclination, the angle observed, 

 instead of 180°, will be only 179° 57' 56": but this error, pro- 

 duced by a defective position of the telescope, is reduced to 

 nothing, if we attend to the making our observation in that 

 part of the field of the telescope where the slightest contact of 

 the objects takes place. 



I shall not speak here of those errors which may result from 

 imperfections of the prisms, since these may easily be calculated, 

 or readily corrected by the artist. Neither shall I say whether 

 the instrument constructed by me, as a first essay, has an- 

 swered the attempt. You, Sir, who have proved it, will be the 

 best judge of it. All that I can say is this: that after having by 

 the use of it ascertained its defects, and the corrections of 

 which il is susceptible, I flatter myself that in executing it 



upon 



