305 



TR ANSCENDEN TAL. 



TRANSFORMATION OF CO-ORDINATES. 



3'16 



it can only be calculated by series. Nevertheless, as tables are now 

 formed of its values, and as many properties and consequences of it 

 are known, it stands in as favourable a position for use as ordinary 

 logarithms at the end of the 17th century. 



TRANSCENDENTAL, a technical term in philosophy, derived 

 from the Latin transcendere, to go beyond a certain boundary. In 

 philosophy transcendental signifies anything which lies beyond the 

 bounds of our experience, or which does not come within the reach oi 

 our senses. It is thus opposed to empirical, which may be applied to 

 all things which lie within our experience. All philosophy, therefore, 

 which carries its investigations beyond the sphere of things that fall 

 under our senses is transcendental, and the term is thus synonymous 

 with metaphysical. Transcendental philosophy may begin with ex- 

 perience, and thence proceed beyond it ; or it may start from ideas 

 (J priori which are in our mind : in the latter case philosophy is 

 purely transcendental ; while in the former it is of a mixed character. 

 [METAPHYSICS.] 



TRANSFORMATION, a general term of mathematics, indicating a 

 change made in the object of a problem or the shape of a formula, in 

 such manner that the orginal problem or formula is more easily solved, 

 calculated, or used after the transformation. Thus it frequently 

 happens that the solution of an equation is facilitated by reducing 

 it to another equation having roots which bear a simple relation to 

 the roots of the former : as an instance, we may refer to the solution 

 of the cubic equation in the article IRREDUCIBLE CASE. 



All the process of algebra consists in transformation, from and after 

 the point at which the problem to be solved is reduced to an equation : 

 so that to write on this subject in detail would require an article on 

 algebra. A few remarks on the leading points which present them- 

 selves in transformations are all we can here undertake to give. 



It frequently happens that transformation points out the nature of a 

 consequence in a manner by which the direct reasoning of algebra is 

 strongly confirmed and illustrated. For instance, when we assert that 

 a quantity has two square roots, one positive and one negative, our 

 assertion is easily verified in its positive part : but it does not follow 

 by the same reasoning that a quantity has only two square roots. We 

 may say that ;e* = 4 is satisfied by x=2, or x= 2, because 2 x 2 = 4, 

 and 2x 2 = 4; but how are we to say that there are no other 

 values which satisfy this equation ? When we transform the equation 

 x" = 4 into (x 2) (x + 2) = 0, with which it is identical, we then see 

 that this product can only vanish when x 2 or x + 2 vanishes ; that is, 

 when a; is + 2 or 2. 



Transformations frequently leave a point unsettled which can only 

 be determined by a subsequent species of experimental test ; or, lest 

 the word experimental as applied to mathematical reasoning should 

 give alarm, by a process of detection which is to choose between 

 alternatives which the process of transformation leaves undecided. 

 This frequently happens when the nature of the transformation is 

 ascertained by means not of the expression to be transformed, but of 

 one of its particular properties. For instance, when we expand a* into 

 a series of powers of x, supposing we proceed upon the property a * x a 

 = a* + ',we find that there is no series fit to fulfil this condition except 



1 + A* + 



2.3 



+ .... 



but we also find that this series is equally fit to fulfil the condition, 

 whatever may be the value of A. So far then our transformation is 

 effected : we see that one among the series formed by giving values to 

 A must be the series we want, if there be any such series. If we make 

 x-\, we then immediately detect the condition which is to give 

 the value of A, namely, that A must be BO taken as to make 



This brings us to the mention of a defect of reasoning which has fre- 

 quently vitiated mathematical works, namely, the assumption of the 

 species of a transformation, and the supposition that only the character 

 of the details remains to be settled, or the individual of the species to 

 be picked out. In the preceding case, for example, it is often stated 

 as follows : " Required the expansion of a* in a series of powers of x." 

 The form of the series is then assumed, say p + r/x + rx 1 +...., and 

 by the use of the property above alluded to, it is found that the series 

 must be of the form I+AJC+ JA*^ + . . . . But all that is here proved 

 is, that if a * be capable of expansion in integer powers of x, the expan- 

 sion must be of the form 1 + \x + . . . . It is true that, looking at what 

 we see in algebra, that science might be strongly suspected to have a 

 peculiar power of rejecting false suppositions, or of indicating their 

 falsehood by refusing to furnish rational results : thus it certainly 

 does generally happen that when we attempt to select from among 

 series of integer powers the one belonging to an expression which 

 really has no such series, we find infinite coefficients, or some other 

 warning. J'.ut it is too much to ask of a beginner that he should take 

 it for granted that algebra has so peculiar a property ; nor, in fact, is 

 it true that such a property is quite universal. It is necessary, 

 therefore, t<> watch all transformations narrowly, both in their general 

 as well as their specific form : first, because there can be no sound 

 reasoning without such caution ; next, because, though it be true that 

 ABT8 AND SCI. DIV. Vol.. VIII. 



in many parts of algebra the science will refuse to acknowledge and 

 obey a false assumption of form, yet it is almost impossible to draw 

 the line at which this refusal ends, and the idea that such a power is 

 universal in algebra will lead the student into many a serious difficulty 

 in the higher branches of mathematics. 



TRANSFORMATION OF CO-ORDINATES. We intend this 

 article purely for reference ; that is, supposing the subject already 

 known, we mean only to put together the formula) in such a manner 

 that any one can be used at once. 



Rectilinear co-ordinates are the only ones which are usually trans- 

 formed ; such a thing rarely, if ever, happens with polar co-ordinates, 

 except in investigations each of which has its peculiar method. And, 

 first, we shall consider rectilinear co-ordinates in one plane, and after- 

 wards in space. What is usually wanted is to express the co-ordinates, 

 of a first system in terms of those of a second, and subsequently given, 

 system. 



And, first, as to co-ordinates in one given plane. 



1. Both systems oblique. Let x and y be the old co-ordinates of a 

 point, x' and y 1 the new ones. Let ju and v be the old co-ordinates of 

 the new origin ; 9 the angle made by the old co-ordinates ; <f> the angle 

 made by the axis of x 1 with jt ; if the angle made by y' with x. Angles 

 are to be measured as explained in the article SIQN ; thus the angle 

 made by x 1 with x means the amount of revolution which would bring 

 the positive part of .r into the direction of the positive part of x', the 

 revolution being made in the positive direction. 



sin (9 < 

 * ~ ** " sin 9 



sin (9 i 



'-' + kr 



sin <p 



2. The old system oblique, tin neio one rectangular. Here if fy is a 

 right angle, and 



sin (9 if>) cos (9 <t>) 



x -M = ~^Tii * - ~~ sinT" * 



sin 9 

 sin <t> 

 sin 9 



COB <t> 



3. The old system rectangular, the nea one oblique. Here, in (1), 9 

 must be a right angle. 



x ft = cos Q . x' + cos i^ . y' 



y v = sin <f> . x' + sin if. . y'. 



4. Both systems rectanyiUar. Here 9 and ty (j> are both right 

 angles. 



x \i. = cos <t> . x' sin $ . if 

 y K = sin <t> . x" + cos $ . ij 



5. The co-ordinatet of the new system parallel to thote of tJte old one. 

 Here 



In any of the preceding cases, if the new and old origin coincide, we 

 have only to make ft = 0, v = 0, and use the formulae accordingly. 



Next, when the co-ordinates are those of points of space. The only 

 two cases which are particularly useful are when both systems are rec- 

 tangular, and when the new one only is oblique. Let x, y, z be the old 

 co-ordinates, and x lt y,, 2, the new ones. Let A, ft, v be the old co- 

 ordinates of the new origin, and let the angle made; by x l and y l be f, 

 that of y, and z, be {, and that of z, and .r, be ij, which we may thus 

 denote : 



Then we have the following formula! : 



x A = cur, !%,+ T-Z,. 

 y /* = o'j-j + /3V, + 7'z, 

 z - v = oV, + /3"y, + 7"*, ; 



Where the meanings of a, $, &c., and the connection of those mean- 

 ings with the places of the letters in the formulae, will be easily caught 

 Erom the following :- - 



Si S* S* 



a = cos x .r, , ft = cos x y, , y = cos x z, 



XV ^ /\ 



a'=cosy.r,, /3'=cosyy, 7' 



/3"=coszy, y'=coszz,. 

 And a, a', &c., are subject to the following six conditions : 

 a' + a' J + a"' =1 $y + 0'y' + P'-y" = 003 { 



&* + ff* + $"* = 1 ya + y'a.' + y'a" = cost) ; 



7 1 + y' 1 + 7"' = 1 a/3 + a'ff + a"0" = cos f 



This case is not much required. The following, in which both sys- 

 tems are rectangular, is of the highest importance. When we speak of 

 the angle made by two axes, we mean, as before, the angle made by the 



x 



