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straight lines enables us to avoid the difficulty connected with our first introduction to the theory and properties of such lines. If we shew, as Playfair has done, that the exterior angles of a triangle are together equal to four right angles, it follows that the interior angles are equal to two right angles: but if the base of a triangle remains finite, and the vertex is removed further and further, the vertical angle becomes less and less, and diminishes without limit; in which case the sum of the base angles is equal to two right angles, and the sides become parallel straight lines; whence the properties of such lines, which are enuntiated in the xxixth proposition of the first book of Euclid, immediately follow.
A good illustration of this theory occurs in the Phenomena of Parallax. If the angles subtended at the centre of the earth by the sun and any fixed star, whose parallax has not been discovered, are observed when the earth is in perihelion and at aphelion, it is found that, notwithstanding the extreme delicacy of our instruments, the sum of these two angles is exactly equal to two right angles. Taking then the two positions of the earth to be the extremities of the base of a triangle, and the line passing through the sun's centre and terminated by them to be the base, and the fixed star to be the vertex, it appears that, although the base of the triangle is 190,000,000 of miles, the angle subtended by it at the vertex is too small to be measured, and the two lines drawn to the star from the earth, at the two positions of it, are to all appearance parallel straight lines.
187.] In corroboration also of what has here been stated, the following are a few out of a great many striking instances: In differentiating tan 6, we have rf.tan 0
which is necessarily a positive quantity; and therefore by Theorem I, Art. 110, 0 and tan 0 are always increasing and decreasing simultaneously, and therefore as 0 increases tan 0 increases. Now as 6 approaches to 90°, tan 0 becomes + oc; and immediately after 0 has passed 90°, tan 6 becomes — oc, indicating that negative infinity is positive infinity increased; that is, as 0 has increased and passed through 90°, tan 0 has increased from + oo to — oo . And so again, as 0 increases from 90° to 180°, tan 0 is continually increasing from — oo to 0, and passes through 0, and increases to 4 oo, which is the value of tan 0 when 0 = 270°; and so on, as 6 increases, tan 0 is continually increasing, travelling, if we may so say, round the circle of which the straight line along which tan 0 lies is conceived to be the limit, when the radius of the circle is infinitely great. It is impossible not to remark how exactly this illustration agrees with what has been said in Chapter I on the order of infinitesimals. For corresponding to every 180° through which 6 turns, tan 0 passes from 0 to oo, and on through sc to 0 again; that is, the path through which tan 6 has travelled is infinite, although 0 has passed over only a finite angle; and therefore, when 0 has revolved througb 360° and 540° and 720°, and so on, tan 0 has travelled over a length of line equal to twice, three times, four times, &c, the infinite length corresponding to a revolution of 6 through 180°; and thus we have infinities bearing a finite ratio to each other. Conceive moreover 0 to have revolved an infinite number of times through 180°, then the distance over which tan 0 will have travelled will be an infinity of infinities, that is, will be (infinity)2; and thus we obtain different orders of infinity.
Again, suppose the following problem to be given: To find the maximum and minimum values of y, when
y = oo, when x — 3; but as ~- does not change its sign, this
value of y is neither a maximum nor a minimum. How then is
the result to be interpreted? As follows: Since ^ is negative,
y decreases as x increases, and when x is a little less than 3, y ~ — oo: but when x is a little greater than 3, y = + oo; therefore, as x has passed through 3, the value of y has changed from — oo to + oo , but y has decreased during this progressive increase of x, therefore + oo is — oc decreased; therefore y has not reached a minimum or a maximum value when x = 3, because it has not become — oc, and then returned, but it has gone on decreasing. And if we draw a graphical representa
tion of the curve corresponding to the equation, such as that in fig. 24, the phenomena explain themselves. The curve on the negative side of the axis of y is of the form Cb, where Ob = 2; and if Oa — 3, the curve is continually approaching the line drawn through A parallel to the axis of y, and when x is nearly 3, y is — oc: but when x is greater than 3, y is + cc; that is, the curve has crossed the asymptote at the pole of the circle of infinite radius opposite to A, and has returned in the direction tr, the branch in the direction of E being a continuation of that in the direction of D. Similarly the branch in the direction r would, if produced, unite itself to that in the direction c, having crossed the axis of .r at the pole opposite to o.
In corroboration of this theory, it will appear that whenever a curve is of the form fig. 24, if the criteria, which will be discussed in the next Chapter, are applied at points such as those where the branch E meets the branch u, and crosses the asymptote, we have the characteristics of a point of inflexion; and if the curve be such as in fig. 25, we have the characteristics of a point of embrassement; and whenever such as is represented in fig. 26, the conditions of a maximum ordinate.
And so again whenever a branch of a curve continues to infinity, it always returns in some way or another; and in whatever manner a rectilineal asymptote is drawn, no branch of the curve ever goes off" asymptotic to it without returning in one of the ways indicated in the figures 24, 25, 26; and it seems impossible to account for such phenomena except on the theory explained above, viz. that the plane and the straight line arc respectively the superior limits of the sphere and the circle, when the radii become infinitely large*.
Section 2.—On the extension of symbols of direction.
188.] In algebraical geometry, and therefore in the applications of the Differential Calculus to the theory of plane curves, we meet with symbols of two distinct characters; symbols of quantity, such as a,b,c, x, y, z, 6,<f>,y\r, when sym
* For a further elucidation of many points in elementary geometry more or less connected with the present subject, I would refer the reader to a small Treatise on the Difficulties of Elementary Geometry, by F.W. Newman, M.A., formerly Fellow of Balliol College, Oxford; Longman and Co., London, 1841. bolical respectively of lines and angles: and symbols of direction, 4- , —, -V — — </( —), &c. Our object is so to enlarge our power of interpreting symbols of this second kind as to comprehend those which are usually called Impossible or Imaginary, of which however we shall discuss only two, viz. + —, — \/ —, or as they may be written, in accordance with the index law, + (-)*, -(-)**.
As to symbols of quantity, it is to be observed that, when we symbolize a line by a, we do not mean that a is the absolute length of the line; for all lengths can only be relative, and there must be some modulus or standard to compare them with: but we intend a line which is in length a times some arbitrary, though for the time fixed, standard unit. So a line symbolized by b is a line b times in length some unit. Thus then a, b are numerical quantities; not concrete magnitudes, but abstract quantuplicities, the subject-matter of arithmetical algebra, and therefore subject to its laws; they do not designate the absolute lengths of lines, but the number of times a certain concrete unit is to be taken. So again if an area is symbolized by a b, a and b are abstract numbers, which must be multiplied together by the laws of arithmetical algebra, and their product is the number of times the superficial unit is to be taken. Let it therefore be carefully borne in mind that this is the meaning of the several symbols of quantity, whether constant or variable, which we shall use in the following Chapters. Suppose that we have a line symbolized by a, and that we fix upon a certain point as the origin from which lines are to be measured, any line drawn from it, equal in length to a times the linear unit, will fulfil the requirements of the single symbol a. But inasmuch as an indefinite number of equal lines may be drawn from any one point, thus far we have no means of determining which of all such lines is intended; hence arises the necessity
* For a fuller explanation of the principles of explaining these and such like symbols, we would refer to Etudes Philosophiques sur la Science du Calcul, par M. F. Valles, 8vo. Paris, 1841; and for the general theory of the
meaning of ( + )« to Dr. Peacock's Algebra, 8vo. Cambridge, vol. i. 1842, vol. ii. 1845; to Mr. Warren's Treatise on the Square Root of Negative Quantities, 8vo. Cambridge, 1828; and to many papers in the Cambridge Mathematical Journal, of which for the most part Mr. Gregory was the author.
of some other symbols to indicate direction, or, as they are called, .symbols of direction or affection. One or two of tho most simple cases of these we proceed to explain, feeling assured that the principle of explanation is so entirely in harmony with the usual meaning of + and — that it ought not to be omitted in an elementary treatise; and also chiefly because it enables us to shew that an algebraical curve, though apparently discontinuous and confined within certain fixed limits, is not in reality so, but extends to infinity in all directions. Other parts of the theory, some of which are as yet not sufficiently established, we omit, as unsuited to our present object.
189 ] Suppose o, fig. 27, to be the point from which lines are to be measured, and Oa = a times the linear unit to be drawn from o towards the right hand. Now since, as we said above, any line drawn from o, a times the linear unit in length, will be symbolized by a, it is necessary to fix on some originating direction; suppose this to be Oa, and any line measured from o towards A to be affected with the symbol of direction +; if then a line, after it has undergone any operation or a series of operations, comes into the position Oa, it is still to be symbolized by + : and, if the line is a, by + a. Such an operation we might conceive to be a reciprocating one, the line at one time being in the position Oa, and at another in the position O A', having moved sideways, and assumed all intermediate positions. Or we may conceive that the line Oa, see fig. 28, has revolved round the point o, and, having turned in the plane of the paper through 360°, has again come into its original position, and so on continually; and it is manifest that as often as it has revolved through any multiple of 360°, it has assumed its original position Oa, and is therefore to be symbolized by + a. So also there are many conceivable ways in which the line may have moved, and that periodically, and at the end of a complete period be in the position Oa. But have we any other customary mode of indicating direction, to serve as a guide which of these conceivable operations to take? We have. Whenever a line equal in length to a is measured from o towards the left, we symbolize it by — a -, if therefore either ( —) were a symbol for the operation of one oscillation having been performed on the line, i. e. the line having passed into the position O'a', see fig. 27: or ( —) symbolized the line Oa, fig. 28,
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