408 Mr. atwood’s Theory for the Menfuratlon 
mutt be lefs than 90°. If BM be lefs than a quadrant, an 
obfervation by two reflections may be taken when the radius 
CP is directed to any point in the circumference of the plane of 
motion. When the arc BM is greater than a quadrant, two 
other limits will be produced in the circumference of the plane 
OCP ; while the radius CP is between thefe limits, no obfer- 
vation by two reflections can be taken : thefe limits are con- 
ftruCted thus (flga 6.). BM being greater than a quadrant, with 
the pole B and diflance BI equal to a quadrant, defcribe the 
arc of a great circle 17 interfering the parallel WM- in the 
points I and i: through I and i draw the fecondaries KY, KZ: 
while the radius CP is between Z, and Y no obfervation can be 
taken by two reflections. If BIE, B i E, be drawn equal to a 
femi-circle, and DE joined, then DE will be the meafure of 
the limiting angle which can be obferved by this confcruction, 
either on one fide of KO or on the other ; and becaufe, by the 
principles of trigonometry, the arcs BD and DE are in the fame 
great circle, BDE being a femi-circle, we fliall derive from the 
conftruction this concluflon : the difference between 180° and 
double the angle of incidence on the fixed fpeculum, will be a 
limit which terminates the angle obferved by two reflections in 
every cafe, when the arc BM is greater than a quadrant 
] 4. In any given example formed on the principles which 
have been demonftrated (fig. 2.) fo.r the efhmation of angles 
by two reflections, three of the four quantities necefiary to 
determine the refult muff confhmtly be the fame, while the? 
fourth, that is, the arc OP, varies with the magnitude of the 
angle fubtended by the objeCfs obferved : the different magni- 
tudes of thefe three given quantities will caufe a great variety 
of properties in conlf ructions which depend on the general 
* This termination cf the angle which can be obferved by two refle&ions may 
happen while the obferved angle is increanng or decreafing during the revolution 
of the index in the plane of motion, 
4 
theory. 
