337 



TRANSVERSE. 



TRAVERSES. 



338 



The projection of figures may throw them into such different forms 

 that lines which, in Euclid's mode of speaking, would be called sides, 

 become diagonals, and rice vend. The distinction of diagonal and 

 side therefore becomes an incumbrance, and a new mode of viewing 

 polygons is introduced, of which we shall now give an instance. A 

 figure contained by four straight lines is, generally speaking, one which 

 has six points : since four straight lines meet two and two in six 

 points : thus, the four-sided figure E F B c has the six points E, r, B, c, D, A. 

 If all these points be joined two and two, we have the additional lines 

 B F, c E, D A, of which only the two first are commonly called diagonals : 

 but all three have common properties. We shall prove the following 

 by the extreme method of projection already alluded to, leaving the 



reader to verify it by the theory of transversals : each of the diagonals 

 is harmonically divided by the other two ; that is, 



AM 



CG 

 BG 



M n : : A N : Nil 

 c E : : c x : N E 

 OF : : BM : MF 



To show the first : project the figure so that N B shall be the 

 vanishing line. Then D E c A will be projected into a' parallelogram, 

 and M G into a parallel to D E and A c passing through the intersection 

 of the diagonals : consequently, in the projection, A M = M D, and we 

 have, also in the projection, 



AM PS _^ . 



ii D ' y A 



for D N and N A are both infinite, and D JJ-=-N A is unity. Now by the 

 test in PROJECTION, which is here satisfied, this proposition must be 

 always true, whence in our figure we have A M : M D : : A N : N i>. 

 Similarly the other proportions may be proved. 



Here then is the easiest way of dividing a line in harmonic propor- 

 tion. Let A D be given, and M : it is required to complete the 

 harmonic division by finding N. Take any point, B, and draw D B, M B, 

 A B. Choose any point, F, in M B, and produce D F and A F to c and E. 

 Then c E produced to meet A D will give the point N required. No 

 instrument is wanted, except the ruler and pencil, and it is a good 

 exercise in drawing to find out by repeated instances that, let B be 

 taken where it may, there is but one position of y. It is also a good 

 test of the straightness of a ruler. 



TRANSVERSE, a name often given to one of the axes of a figure, 

 usually that of greatest magnitude or which goes most directly across 

 the figure. Thus the longer axis of an ellipse or hyperbola is called 

 the transverse axis: but sometimes the shorter axis is so called. 

 Properly speaking, it ought to be only a term of relative distinction : 

 either axis is transverse to the other. 



TRAPE'ZIUM, TRAPEZOID. The first word (rpawcfav, a little 

 table) is used by Euclid for, or at least denned to be, any four-sided 

 figure which is not a parallelogram. The second word, formed from 

 the first, has been used by various writers, and in different senses. A 

 trapezoid, says Harris (' Lex. Tech.') is a solid irregular figure, having 

 four sides not parallel to one another : Hutton repeats this, but says 

 it sometimes means a trapezium, two (only) of whose sides are parallel 

 to each other. What the solid figure with four sidet means we do not 

 know : but as the word is never used, we omit all inquiry about it. 

 Words however are so scarce in mathematical language, that it is a 

 pity when any become obsolete. If we were to sugge.it meanings for 

 these terms, we should propose that trapezium should be the general 

 word for plane four-sided figures, parallelograms and all ; and that 

 trapezoid should denote a four-sided figure whose sides are not in the 

 same plane. Perhaps this is what was intended by the tulid figure of 

 four sides : if so, it was particularly unnecessary to state that the 

 sides are not parallel. 



TRAVKUSK, in Law, is a contradiction of some matter of fact 

 alleged in pleading by the opposite party. Generally all matter of fact, 

 that is material, ought to be either confessed and avoided, or traversed ; 

 and if a party justifies an act as to one particular time and place, or 

 confesses and avoids in one respect, he ought to traverse it as to all 

 other. Otherwise what is materially alleged will be taken to be 

 admitted. Traverse of an immaterial fact, or of a mere supposition, 

 or of inducement, is bad, for it is not an answer to the action. If a 



ARTS ABD SCI. DIV. VOL. VIII. 



traverse is tendered as to a material point by one party, the other 

 must accept it ; he cannot waive it and tender another traverse. 



In criminal pleading traverse of an indictment is the taking issue 

 and denying some material point in it. Not guilty is a general 

 traverse, which throws on the prosecutor the necessity of proving all 

 the material facts. 



TRAVERSE SAILING. [RECKONINGS AT SEA.] 



TRAVERSE TABLE. In navigation two tables bear this name ; 

 the one is the list of courses, distances, northings, southings, eastings, 

 and westings, in which they are arranged for the convenience of 

 addition, so that difference of latitude and departure may be readily 

 deduced in dead reckoning, when a ship cannot lay her course, but sails 

 in various directions, or upon a traverse as it is called. [RECKONINGS 

 AT SEA.] That which is more properly called a Traverse Table, is 

 one from which the various northings, southings, &c. as above are 

 gathered. 



The following figure will explain its principles : 







Suppose a ship at c sails S. 30 \V., a distance of 231 miles ; accord- 

 ing to the usual mode of construction, the side c B would represent the 

 meridian of the ship, and, therefore, the distance c B would be the 

 difference of latitude ; while A B would be the departure from that 

 meridian, and A c the distance sailed, the angle c being the course 30, 

 as marked. It will be seen that only four parts enter into considera- 

 tion, because the angle A is known (as the complement) when c is 

 known, and the L B is a right angle : any two of these four parts 

 being known, the others may be found by trigonometry. But to save 

 the trouble of logarithmic calculation (which such involves), a large 

 number of right-angled triangles have been purposely computed, and 

 varied according to the magnitude of the course or Z c : being 

 arranged in columns headed by the amount of such angles : the above 

 example would appear in the table thus among the distances : 



DUt. 

 231 



Course 30. 



Diff. Lat. 



2001 



Dep. 

 115-5 



Comparing this with the figure, the nature of the table will be evident, 

 as each line represents three parts of a triangle, as taken in connection 

 with whatever Z. c heads the list or page. 



The artisan and engineer would be greatly benefited if these tables 

 were more in use. As it is, they are too generally considered as adapted 

 to navigation principally, and the terms used favour this misapprehen- 

 sion. If, however, such tables were published with the substitution 

 of / c for " course ;" perpendicular for "difference of latitude," and 

 bate for "departure," the difficulty would be at once removed. Such 

 tables supersede calculation of right-angled triangles entirely, so far as 

 application to practice demands. How far the facility thus offered 

 would tend towards a superficial standard of mental attainment is 

 another consideration : but such are the capabilities of the traverse 

 table. 



TRAVERSES, in Fortification, are usually masses of earth which 

 are raised at intervals across the terreplein of a rampart or across the 

 covered-way of a fortress : then- positions in the covered way are indi- 

 cated at t, t, &c. [FORTIFICATION.] On a rampart they serve to pro- 

 tect the guns and men against the effects of a ricocheting or enfilading 

 fire, which might otherwise dismount the former, and compel the latter 

 to abandon the parapet ; and in the covered-way, besides serving for 

 similar purposes, they constitute retrenchments behind which the de- 

 fenders may keep up an annoying fire of musketry upon the enemy, 

 should the latter attempt to force his way along the branches of that 

 work. On this account they are provided with banquettes, or steps, 

 on which the defenders may stand to fire over them. Such a work, 

 when formed in a direction parallel to a rampart or parapet, on its 

 interior side, for the purpose of securing the defenders against a fire 

 from the ground in their rear, is called a parados: 



Palisades ar planted along the banquettes, in order to prevent the 

 assailants from suddenly passing over the traverses ; and, at the passage 



z 



