Formulæ for the

934. Again, the ordinary formulæ for the solution of triangles solution of sometimes give results, which are less accurate than those which

fer little

triangles are commonly obtained, when they involve the cosines of angles which involve the which differ little from zero or 180°, or the sines of those which cosines of angles differ little from 90°, inasmuch as, under such circumstances, which dif- their values change slowly for considerable changes in the value of the angle: the series, however, for the sines and cosines of the sines of angles, will enable us to deduce formula which are adapted to such extreme cases, and where the minutest changes of value in the quantities sought to be determined will become immediately sensible.

from zero, or 180o, or

those which differ little from 90o.

Thus, in a triangle, where A and B are very small and where C is very nearly 180o, we get (Art. 885), if π – 0 be the measure of C,

omitting all terms of the series for cos 0 beyond the second, as being too small to affect the result within the limits of the recorded places of figures; by extracting the square root, we get

Again, if a and ẞ be the measures of A and B*, we get

When we speak of the sines and cosines of angles, it is indifferent whether the angles are expressed by their measures, or by degrees and minutes: but it should always be kept in mind that the numerical values of the measusres of angles enter essentially into the series which we are now considering.

a very convenient and very accurate formula for expressing one side of a triangle in terms of the two others and of the measures of the adjacent angles, provided those angles are small.

CHAPTER XXXVIII.

indefinite.

ON THE RELATIONS OF SERO AND INFINITY.

Meaning of 935. THE terms infinite and indefinite are frequently used the terms infinite and indiscriminately by mathematical writers, though, if due regard was paid to propriety of language, they should be distinguished from each other: they are negative terms whose meaning must be defined by that of the terms finite and definite, which are respectively opposed to them.

Infinity and

zero.

A finite number, a finite line, a finite space, a finite time, would denote any number, line, space, or time, which is either assigned or assignable: whilst the term definite could properly be applied to such of those quantities only as were already assigned or determined: in other words, the term finite is more comprehensive than definite, being limited only by the power possessed by the mind of conceiving the relations which the magnitudes, to which it is applied, bear to other magnitudes of the same kind.

An infinite number, an infinite line, an infinite space, an infinite time bear no conceivable or expressible relation to a finite number, a finite line, a finite space, or a finite time: the term indefinite, properly speaking, when applied to these quantities, would imply nothing more than that they were not determined or not assignable.

936. Magnitudes may be infinitely small as well as infinitely great, and the abstract term infinity should be, properly speaking, equally applicable to both, though it is confined, by the usage of language, exclusively to the latter, whilst the term zero is exclusively applied to the former: the general term infinity is superseded by the specific terms immensity and eternity in the case of space and time*.

The phrase for ever, though properly expressing infinite duration of time, is commonly applied to denote infinite repetition as well as infinite time: thus the processes which never terminate are said to be continued for ever: the terms

937. The symbol is used to denote magnitudes which Symbols of infinity and are infinitely great, in the same manner that the symbol 0 is used to denote those which are infinitely small: they are con

a

a

nected by the equation == ∞, and 0, where a may be a

0

finite though indeterminate magnitude: in the first case, we consider as the quotient of the division of a by 0: in the second, we consider 0 as the quotient of a divided by ; such results may be interpreted by considering the dividend as the product of the divisor and quotient: thus, there is no finite number which, when multiplied into zero or an infinitely small number (fractional or decimal) will produce a finite product: there is no finite line, which multiplied into zero or an infinitely small line, will produce a finite area, and similarly in all other cases.

zero.

orders of

938. The product of ∞ into 0, or of an infinitely great and Different an infinitely small number, line, or other magnitude, in the sense infinities which we have attached to those symbols and terms, may produce a finite result, but it does not follow that it must do so: symbol.

a

the equation = ∞ is universally true, when a is finite: but the same equation is also true, when a is infinite, and therefore

∞

replaced by in other words ∞ in such a case, the

0

infinity denoted by the symbol ∞, on one side of the equation, is said to be infinitely greater than the infinity denoted by the same symbol on the other: for one is equivalent to, and the

a

∞

other to, and the ratio of the second to the first, or X

denoted by the same

tion of the

939. The mind is as incapable of conceiving the relation The rela of different orders of infinities as it is of conceiving infinity infinities itself, and it is only when the relation between them is the the same necessary result of symbolical language, and of those general symbol ∞

denoted by

may be

come definite when

of a series which are said to be continued in infinitum, are also said to be con- the circumtinued for ever: but it should be observed, that the notion of infinity of time stances of is closely associated in the mind with all our notions of indefinite repetition.

their origin are known.

Example.

laws of their combination, which the rules of Algebra impose upon them, that they can become the proper object of our reasonings for the same symbol is used equally to denote all magnitudes which are infinitely great, and the same symbol O to denote all magnitudes which are infinitely small but if the symbols ∞ or O were used, in a course of operations as ordinary symbols, when the circumstances of their usage shewed a common origin, and therefore indicated something beyond a mere symbolical identity, we must adopt the results of operations upon them, whether finite or not, in the same manner as any other necessary results of Algebra: thus, if the symbols and denoted the same infinite magnitude (such as

when = 1), the relation of ax∞ and b× ∞,

a

2

1-'

would

be equally as if, in this ratio, had been replaced by an ordinary symbol of algebra: but, if the course of our reasonings should call our attention to a x ∞ and b x ∞, as simply denoting two infinite magnitudes, without any reference to the relation which the circumstances, in which they originated, may make them bear to each other, we might properly represent a × ∞ and b× by the common symbol ∞, and the relation between a×∞ and b×∞ would become altogether indeterminate.

But though the symbol might not have the same origin in the expressions ax∞ and b× ∞, yet if the precise symbolical conditions of their origin in both cases were known, the indetermination of their relation to each other might be removed: thus, if the symbol in ax∞ originated in the expression 2 when x=1, and the symbol in bx ∞ originated in the

1.

-X