KNOWLEDGE AND FAITH. 507 
the sum of the angles shall be nothing, another in which it shall be equal to six right 
angles,* and others in which it shall be equal to any quantity we please, from 0° to 540°.” 
182. This conversation and experiment would not weaken the belief in the truth of the 
original proposition, as it had always been understood. It would merely show that the 
judgment had assumed too much, or that the definition of the term “triangle” was too 
limited, and that the proposition was true only of rectilinear plane triangles. It would 
also show that demonstrable truth may lead us into error, if it is not perfectly understood, 
and if its full extent and limits are not properly recognized. Hence the seeming paradox, 
that a thing may be proved true, though it is absolutely false.f 
183. Of a similar character are the errors which we attribute to the senses. The 
nerves connected with each organ of sense are designed to convey certain appropriate 
* This statement is true only in the sense in which the Calculus disregards differentials. In order that there may 
be an angle at the junction of any two sides of a spherical triangle, each of the three angles must be less than 
180°, but it may differ from 180° by a quantity less than any assignable value, therefore it may virtually be re- 
garded as equal to 180°. The algebraical fallacy in the following note, shows that it is not always safe to disre- 
gard differentials. 
+ The following algebraical demonstration that 1 is equal to 8 will illustrate my meaning. 
Let z=a 
Then x? —2ax + a? =a? — 207 + @=2?—a? 
(a@—a) («—a)= (@#—a) (« + a) 
x—a=a+a 
z=a2+ 2a= 32 
= 8 
The error in this case consists in assuming that any factor which has no value, can be used as a factor in deter- 
mining the numerical value of a quotient. 
Peter Barlow (Hlementary Investigation of the Theory of Numbers, London, 1811, Prop. IX), demonstrates 
that ‘the sum of any number of prime numbers in arithmetical proportion, is a composite number.” He over- 
looked the arithmetical series 1, 2, 3, in which 1 + 2=8, and 2 + 3 = 5,—hoth 3 and 5 being prime numbers. 
With this exception, the demonstration is perfectly rigorous. 
Prof. Pierce (Mathematical Monthly, October, 1858), gives a number of “ Propositions on the Distribution of 
Points on a Line,” all of which are rigorously true in their intended meaning; but in some of the cases, it is 
necessary either to suppose that the line is straight, or that the distances between the assumed points are mea- 
sured on the line, and not in the direction of one point from the other. 
Such instances in the “exact” sciences, teach the necessity of a precise understanding and exposition of the 
principles, as well as of all the relations involved in any train of reasoning. The very possibility of error is a 
proof both of liberty and of imperfection. 
We can reason only about that which we can define, and we can define any proposition only as it is comprehen- 
sible to us. All contradictions and errors can probably be traced to errors of definition. It may often be seen by 
impartial observers, that two disputants are both right, and that they differ only because each does not see the 
phase of truth at which the other looks. 
