188 REPORT—185/7. 
On Infinite Angles, and on the Principle of Mean Values. 
The exemplification of these subjects by the foregoing theory depends mainly 
on the distiaction above pointed out, between considering a position merely 
as a point reached, and having one fixed relation with the origin, and consi- 
dering it with reference to the course by which it has been reached. The 
former mode of consideration is merely a limiting view of the latter. If we 
conceive a point on a sphere starting from the origin of longitude and tra- 
versing an unvarying course inclined to the meridian, and if we at any mo- 
ment inquire, first, what amount of longitude has it traversed; and secondly, 
in what longitude is it now posited, the answers to the two questions must 
obviously be, in an algebraic sense, the same ; and in a geometrical sense also 
they must be the same wherever they are intelligible, that is, for every po- 
sition from the equator up to and exclusive of the pole. The assertion of 
this geometrical proposition (the identity of the answers to the two queries), 
extended by the substitution of tnclusive for exclusive, does not involve any 
principle other than the axiom, that, what is quantitatively true up to the 
limit, is quantitatively true at the limit; for in this case the passage from ‘ up 
to the limit’ to ‘at the limit,’ may be considered in such a light as not of 
necessity to involve any change in the character of the subjects or ideas with 
which we are dealing, or to transfer our conceptions from calculable magni- 
tude to something no longer the subject of calculation. The passage from 
the equator of a sphere to the pole by an unvarying course does not of 
necessity involve the consideration of infinite magnitude, for the linear space 
traversed in getting up to the pole is Saad where a is infinitely small; and 
the linear space traversed in actually attaining the pole is > = ¢ both of 
1 
which are finite and calculable. It is true that an infinite amount of angular 
longitude is traversed, but this consideration does not interfere with the cer- 
tainty of our actually attaining the pole in a finite time at a finite rate of 
progress ; and there is nothing in the geometrical character of the problem 
which could lead us to believe that the above inquiries are of totally different 
natures when applied to the pole, and when applied to a point indefinitely 
near to the pole. Let us then consider the point as having reached the pole 
in this manner, and propose the two inquiries above suggested. 
If we ask, what amount of longitude has been traversed ; the answer is, an 
infinite amount. If we ask the geometrical question, in what longitude does 
the point now exist; the answer is, that it is in every possible longitude 
throughout the whole cycle of longitude. 
We are thus led by these considerations to the inference that the idea (as 
applied to geometry) of an angle which in an analytical sense is infinite, and 
the idea of an angle which has at one and the same time every real magnitude, 
are one and the same idea. An angle in geometry, when made to vary by a 
uniform process, is of necessity periodic in its magnitude ; but when the sym- 
bol representing an angle is imported into algebra by the introduction of its 
trigonometrical functions into general analysis, then the angle or its symbol 
must of necessity be considered as having a progressive magnitude, and as 
being capable of having every real value from negative infinity to positive 
infinity ; and the foregoing considerations tend to the conclusion, that an in- 
finite angle, in the latter sense, is the same thing as that angle in the former 
sense, which has at the same time and in one conception every real value ; 
an idea, the perception of which is facilitated by the circumstance that we 
can geometrically depict a position whose angular distance in longitude from 
fx é 
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