TRANSPOSITION. 



TRANSVERSAL. 



other writer* on the theory of punishment, have condemned the general 

 principle of transportation ; and comparatively little has been urged in 

 opposition to their argument*. Mr. Bentham's objections will be found 

 in a chapter on Transportation, in his ' Theory of Punishment* ; ' the 

 archbiahop of Dublin 1 *, in hi two Letters to Earl Grey. 1 These argu- 

 ment*, but Ktill more the resolutions of the colonies not to receive any 

 mom convict*, led to the abandonment of the system. The colonist* 

 were no doubt right in refusing to let their society continue to be 

 the receptacle of the crime and profligacy of the parent state ; but it 

 is not so clear from doubt that in an early stage of settlement the 

 system of transportation may not be adopted with advantage ; indeed 

 the remarkable prosperity of Australia, if not its very existence as a 

 settlement, is owing to this system. Where the population is small, 

 and labour scarce, the criminal is removed from much temptation, 

 and placed in the very best position for retrieving his character ; 

 while the settler has the benefit of cheap and constant labour. The 

 expense, however, to the parent state is large ; it was estimated that 

 in transporting to Australia each convict cost the state 821. 



TRANSPOSITION, the name given in Algebra to the process of 

 removing a term from one side of an equation to another, changing its 

 ign. Thus, if o 6 + r, by transposition, a e= b. On this we have 

 only to remark, that in this instance the rule is not much of an abbre- 

 viation. If we say " transpose c," instead of " take r. from both sides," 

 so little is gained, that it may be doubted whether it would not be 

 better to follow the Continental writers in the use of the latter form 

 of expression : a process which would have the advantage of being a 

 perpetual appeal to reason instead of rule. 



TRANSPOSITION, in Music, is a change of the original key to one 

 higher or lower. This is generally performed at a moment's notice by 

 the accompanist to suit the convenience of the singer. To the latter, 

 transposition is not attended with any difficulty : the change is little 

 more than imaginary, except so far as relates to the compass of the 

 voice. To the accompanist it is far otherwise ; unless playing from 

 memory, he must assign to all the notes as regards their pitch or their 

 situations on his instrument, names wholly different from those in the 

 copy placed before him. To accomplish this he has to tuppoK a change 

 of clef, or clefs, and thus give new designations to all the lines and 

 spaces. For instance and without going into the extreme case of 

 transposing from a score a pianoforte player is required to transpose 

 an air a whole tone lower, from A to O. For this purpose he must 

 assume a change in both clefs, the treble into the tenor, and each note 

 to be played an octave higher than it is written ; the bass into the alto, 

 and each note to be played an octave lower than it is written. Example 

 in A. 



p 



. 



The same as read by the performer, when transposed to Q : 

 8v higher 





I 



8vc lower - 



The difficulty attending this process is so great, that no amateur, 

 and few discreet musicians, unless professed accompanists, or well 

 acquainted with the composition to be transposed, will undertake the 

 tai-k ; for to perform it in an artist-like manner, at first sight, requires 

 a degree of practical skill only to be gained at the expense of much 

 time that might be employed to far greater advantage in studying 

 those higher branches of the art in which the most experienced will 

 always find something to learn. Our remarks, it will be understood, 

 relate to performers on the pianoforte and organ. To those who read 

 from a single staff, and play single notes only, as violinists, flutists, &c., 

 the task of transposing is comparatively easy. 



To meet all the demands of transposition, a familiar knowledge of 

 no lest than seven clefs is necessary, and two of these the mezzo- 

 soprano, and baritone, or bass clef on the 3rd line [CLzr] may be 

 said to have become obsolete. 



The annexed table will exemplify the use of clefs in transposition. 

 It shows bow to transpose a given key-note A for instance into any 

 other note of the scale, and, consequently, how to transpose the whole 

 of any composition. It is hardly necessary to add, that the eemitonlc 

 scale, as concerns line and space, is governed by the diatonic; that 

 A :. A>, 4c., have the same places in the staff as the natural notes 

 represented by the same letters. 



A B C OK V Q 



THAN TIATION. [SACHAIIMTS]. 



TRANSVKUSAL, a name lately given to a line which is drawn 



across several others, so as to cut them all. The word is used in this 



in the writing* of Caruot, Poncelet, Chasles, &c. 

 Let there be a triangle ABC (the reader may easily draw the figures 

 of this article fur hinuelf) and let a transversal cut its three side* 

 Internally or externally ; namely, let A B cut the transversal in r, B c in 

 a, and c A in ft. Then will either one or three of a, ft, t, be in sides 

 produced, and 



AcxBaxcftBcxcax Aft; 



and the converse, namely, if a, ft, e, be points taken on the three sides, 

 having either one or three external, for which the above relation is 

 true, then those three point* are in the same straight line. In the 

 language of Euclid, the ratio compounded of the three ratios of A c to 

 B ft, B a to c a, and c ft to A ft, is the ratio of a line to its equal. 



This proposition is now frequently demonstrated in elementary 

 works on geometry as follows : From any one of the vertical points, 

 A, B, c, draw a parallel to the side of the triangle which does not pass 

 through that point; from c, for instance, draw CM parallel to A B, 

 cutting the transversal in M. Then we have two pairs of similar 

 triangles, M c ft, c A ft, and o B c, a c M, which give 



ac : CM : : OB : Be 



CM : eft : : AC : Aft 



or ca : eft : :ACXBO :A&XBC; 



whence the proposition required is obvious. The converse readily 

 follows by indirect demonstration. 



Let any point P be taken inside or outside of the triangle ABC, and 

 let A p, B p, c p, cut B c, c A, A B in a, ft, e. Then cither one or three of 

 the points as a, ft, c, are internal. 



ACXBaXCft = BOXCftXA, 



which is proved by using the former property in the triangle A B a 

 with the transversal c p c, and the triangle AOC with the transversal 

 Bpft. The converse is also easily proved, namely, that if a, ft, c be 

 so taken on the sides with one or three internal as to satisfy the above 

 relation, then A a, B ft, c c all meet in one point. 



The same proposition as the first is true of any polygon whatso- 

 ever : thus, let ABODE be a pentagon, the sides of which are cut 

 externally or internally by a line, namely, A B in e, B c in d, c D in r, 

 DE in a, EA in ft; then 



A'I. ji r. cd. Tie. 



AO. Be?, cc. Da. Eft. 



For let B D and B E meet the transversal in m and n : then the three 

 triangles BOO, BED, B A E arc cut by a transversal, giving 



Bm.De.co'=Dni.ce.Ba > 

 Di.Ea.nn = Bm.Da.Eii 

 En . A* . BC=BB. Eft. AC; 



by multiplication of which the theorem follows. 



If the transversal be parallel to either of the sides, the two segments, 

 which are then infinite, are to be considered as equal, and removed 

 from both sides of the equation. 



In the article PROJECTION a test is given which, being satisfied, 

 shows that a proposition is true of any figure, if it be true of any one 

 of it* projections. This test is satisfied in all the preceding case*, so 

 that it is enough to prove any one case of these propositions, that is, 

 for any one projection of the figure. Now there is no case in which 

 they are obviously true, A priori, except for that projection in which 

 the transversal become* the vanishing line, or all the segments become 

 infinite. If we put the first proposition of all in this form 



A Bj C_6_j 

 A(T Be' CO 



it is obviously true when the line abe is at an infinite distance, each 

 of the ratios being then unity. It would not be safe, upon the proof 

 given in the article cited, to allow this extreme case of projection to 

 enter into the theorem : nevertheless, other proof might be given, 

 which would make this very simple and perceptible instance, the 

 truth of which is seen at once, sufficient evidence of all the others. 

 We mention this only to show the very great power of the geometry 

 of projections : our limits do not allow of our entering further into the 

 subject. 



The theory of transversals may be made useful in surveying, par- 

 ticularly in military surveying : as an instance take the following. 

 There is an inaccessible point A, from which to B it is required to find 

 the distance without any instruments except signal-poles and a 

 measuring-line. At B set up a signal, and another, c, at a com 

 distance between B and A. Choose another signal-point, D, and 

 between D and B set up a signal at K, and another at F, between D and 

 c, and also between E and A. All this must be done by trial. Then 

 measure D K, E B, D r, r c, and B c. The triangle D B c, cut by the trans- 

 venal E r A, gives the following relation : 



DR. BA . CF = EB . CA.FD 

 OrDE.Cr.BA = EB.FD (BA BC) 

 B . >P . BC 



whence BA= EB. ID-PI, cr 



