Wrights Mathematical Papers. 79 
pendiculars, these evanescent quantities must be brought into the 
calculation, and expressed by 0 
When one number is either added to another, or subtracted from 
it, the result is called a factum. 
Place A before, and after, the evanescent angle of commutation 
belonging to the first side. ‘Then, the several angles of commuta- 
tion may be found by the following rule, which contains two cases. 
Case I. When the letters, placed before, and after, the angle of 
commutation last found, are the same. 
Take the sum of the angle of commutation last found and the given 
angle, if it be inward; but if it be outward, take their difference. 
Case Il. When the letters, placed before, and after, the angle of 
commutation last found, are different. 
Take the difference of the angle of commutation last found and 
the given angle, if the given angle be inward ; but if it be outward, 
take their sum. : 
Tn both cases, when the factum exceeds 90°, subtract it from 180°; 
but when it exceeds 180°, subtract 180° from it, and the remainder 
will be the required angle of commutation. 
The position of the perpendiculars and bases may be discovered 
by inspection ; thus, 
Case I. When the factum, last obtained, is less than 90°. 
Prop. 1. If the angle of commutation be subtracted from the given 
angle, the letters placed before the two angles of commutation last 
found will be unlike, and the letters placed after, like. 
Prop. 2. If the angle of commutation is not subtracted from the 
given angle, the letters, placed before and after, will be unlike. 
Case I. When the factum, last obtained, is greater than 90°. 
Prop. 1. If the sum of the angle of commutation and given angle 
be subtracted from 180°; the letters placed before will be like, and 
the letters placed after, unlike. 
Prop. 2. If the sum of the angle of commutation and given angle - 
is not subtracted from 180°; the letters, before and after, will be like. 
When the factum is 90°, it may be considered as belonging either 
to Case I. or Case II.; and when the factum is 180°, it may be con- 
sidered as belonging either to Prop. 1 or Prop. 2, Case II, each sup- 
position leading to a true result. 
From a traverse table, find the bases and perpendiculars of the 
several sides. Put the bases underneath the sides in the second 
column, and the perpendiculars in the third column. Prefix N, or 
the negative sign, to those perpendiculars whose positions are desig- 
