TRAJECTORY. 



TRANSCENDENTAL. 



I i 



tragaoanth of Dic*ooride*. The A. tmgara*tlia (Tar. a. Linn.), the X. 

 ttmadteiitit (Lamarck et Doc.), long reputed to be the noun* of tri- 

 gBoanth, yield* no concrete gum, but merely a gummy juice, which is 

 tuod in confectionary. The A. Dicktmii (a name substituted by Dr. 

 Boyle fur A. Xnbilifena of Lindley) yields the reddish-, 

 tngacanth. In the hot months of July and August, particularly after 

 a dewy or a cloudy night, the branches of A. rmu are found encru-t- il 

 with tragacanth. It is not procured by artificial incisions, but exudea 

 spontaneously from natural clefts in the bark, or from punctures made 

 by insects, or more probably by a subepidermal fungus, like the nema- 

 tpora rrocra, as the ahrubs from which the juice exudes are always in 

 an unhealthy state, or ready to perish. (Decondolle, ' Phys. Veg.,' 

 Li). 174.) 



In commerce tragacanth occurs in two forms, termed vermiform, 

 nd flake or cake tragacanth. The former, called also Morta tragaeanth , 

 is not frequent in this country. It is mostly in small twisted thread- 

 like pieces, seldom in Bat or bandlike portions, of a variable size, of a 

 whitish colour. The larger irregular pieces often run together, and are 

 of a yellow or yellowish-brown colour. White worm-like pieces are 

 selected and sold as vermicelli. Flake or Smyrna tragacanth occurs in 

 tolerably large, broad, thin pieces, with concentric elevations or lines, 

 seldom of a filiform shape : colour whitish. Both sorts are hard, yet 

 somewhat soft and even flexible before breaking ; fracture dull and 

 splintery. It is with difficulty reduced to powder, except in winter, 

 or in a heated mortar. It is devoid of taste and smell. It swells in 

 the mouth, and is lubricous. Fine tragacanth is not rendered blue by 

 iodine, but the Morra tragacanth is affected by it, as well as an artificial 

 substance prepared by boiling starch, which last article, called traga- 

 cautin, does nut swell in water. Kutera gum, the produce of a species 

 of cochlospernium and sterculia, which is sometimes mixed with or sub- 

 stituted for genuine tragacanth, is not affected by tincture of iodine. It 

 always occurs in stalactite-like pieces, and consisting almost entirely of 

 bueorin, is scarcely soluble in water. [OUM ] 



Tragacanth approximates more to starch than common gum, than 

 which it is more nourishing, but less digestible. Tragacanth is to be 

 preferred to gum-arabic to form a mucilage, as one part will inspissate 

 fifty part* of water. It is better to allow pieces of tragacanth slowly 

 to dissolve in cold water than to use the powder with boiling water. 

 Both the mucilage and powder are used to suspend heavy powders in 

 water ; also to make lozenges and pills. For electuaries it is improper, 

 as it renders them slimy on keeping. As a demulcent, or means of 

 sheathing the fauces and intestines, it is preferable to gum-arabic, its 

 insolubility rendering it a more efficient protection to the mucous 

 membrane against either acrid poisons or unhealthy secretions. Thus 

 in India, tragacanth bulled in rice-water is advantageously administered 

 in dysentery and bloody fluxes. Externally, a thick mucilage of tra- 

 gacanth is a good application to burns, to exclude the air. 



TRAJECTORY, the technical name which was formerly given to a 

 curve, that is, to a curve required to be found by means of given con- 

 ditions ; most frequently used for the required path of a particle acted 

 on by given forces. The term is now seldom used. 



TRAMMELS, the name of the ELLIPTIC COMPASSES, described in 

 that article, in which a bar carrying a pencil is guided by two pins 

 which move in grooves. 



TRAMWAY. A track laid down on the surface of a common road, 

 for the purpose of diminishing the effort of traction required in moving 

 wheeled vehicles ; and fur this purpose it is necessary that the material 

 of the tramway should be practically incompressible, and as smooth as 

 possible. Iron, wood, marble, and granite are used in the formation of 

 tramways, as in the cases of Train's Street Railways, the forest roads 

 in wood countries, in the Italian cities, and in the Commercial Road, 

 London Docks. See RAILWAYS ; TRACTION. 



TRANSCENDENTAL, a mathematical term of description, the 

 meaning of which is not very uniform. When any particular formula 

 is incapable of being expressed by any particular range of algebraical 

 symbols, it is, with respect to those symbols, transcendental that is, 

 it transcends or climbs beyond the power of those symbols. The word 

 ww perhaps first used by Leibnitz (' Leipzig Acts,' 1686), who says, 

 " placet hoc loco, ut magis prpfutura dicamus, fonttm apertre trcmscen- 

 deatium r/uanli>atum, cur nimirum quecdam problemata neque sint 

 plana, neque solida, neque sursolida, aut ullius certi gradus, Bed omnem 

 oquationem algebraicam transcendant" Here, then, is the first 

 meaning of the word ; a transcendental problem is one the equation 

 of which is infinitely high, or contains an infinite series of powers of an 

 unknown quantity, so that its highest degree transcends every degree. 



TO form an idea of what u now most commonly me.int by transcen- 

 dental, it will be desirable to recapitulate the steps by which algebra 

 has arrived at its present state of expression, or, rather, mathematical 

 analysis, u those would say who do not like to call the differential 

 calculus by the name of algebra. 



And first we have the state which preceded the time of Vieta, in 

 which formula! were mostly described in words, and the adoption of 

 arbitrary symbols of quantity was only of casual occurrence. 



Not, we have the introduction of arbitrary symbols of quantity by 

 Vieta, but not to the extent of using arbitrary numbers of multipli- 

 cations, or algebraical exponents. Here what we now call a" was 

 transcendental ; Vieta could have described o by a cubo-cubum, or a' 

 by a quadrto-quadrato-cubum, but o had neither name nor symbol 



Thirdly, we have the stage which began with Harriot and Descartes, 

 and which brought ordinary algebra into substantially ita present form. 

 During these periods, however, geometry and arithmetic, without help 

 from algebra, had brought into use sines, cosines, &c., and logarithms, 

 which were then properly transcendental. The wonls which described 

 a particular mode of drawing lines in a circle, or the result of many 

 interpositions of geometrical means between two given numlx 

 not place those lines or means among the objects of algebra, and gave 

 no clue to any algebraical properties. 



Fourthly, we have the short but interesting period in which, before 

 the formal invention of fluxions or the differential calculus, infinite 

 series began to be employed, and the transcendental* last alluded to 

 ceased to be absolutely incapable of expression. This was the state in 

 which Leibnitz found the science when he first proposed to distinguish 

 between algebraical and transcendental problems. 



Fifthly, we have the period succeeding the invention of the diffe- 

 rential calculus, in which the areas and lengths, 4c., of curves could be 

 expressed, whether they could be reduced into older language or nut, 

 by the new signs for fluents or integrals. 



Sixthly, we have an alteration which it might have been supposed 

 should have come long before, namely, the expression of the old tr.m- 

 scendentals as recognised functions, and the writing of them accordingly, 

 as log x, sin x, cos x, &o. Strange as it may appear, this was never 

 done till the time of Euler. And it is only in our own day that the 

 system has been completed by the recognition of the number whose 

 logarithm is x, the angle whose sine is x, &c., as functions of .r, and 

 the adoption of the appropriate symbols log' x, sin~' x, Ac. 



Seventhly, a most important addition has been coining into use in 

 the present century, namely, the employment of definite integrals as 

 modes of expression, not merely of functions of the variable of integra- 

 tion, but of other quantities which only enter ag constants, or wlii. -h, 

 if they vary, vary independently of the variable used in integration. 

 So powerful is this mode of expression, that it may almost be sus- 

 pected to be final ; and the word transcendental is rapidly acquiring a 

 new meaning. We predict that it will settle into the following : a 

 transcendental result will be one which is incapable of expression 

 except by a definite integral, or by an infinite series which cannot be 

 otherwise expressed than by a definite integral. 



In the meanwhile there are two senses in which the word is used. 

 The first is that just explained ; the second has reference to the old 

 distinction of algebraical and transcendental. A function of x is alge- 

 braical when it is finite in form, and x U never seen, nor any fun t !! 

 of it, in an exponent, nor under the symbols of a sine, cosine, &c., or a 

 logarithm. No operation then enters with x unless it be one of tho 

 four great operations of arithmetic, or else involution or evolution with 

 a definite exponent. Thus, in this sense of the word, log x and sin x 

 are both transcendentals. But in the modem sense in which transcen- 

 dental is not opposed to algebraical, but to that which is expressible 

 by ordinary means, log x and sin x are not transcendental, being among 

 the most common of the present modes uf expression, and being, in fact, 

 connected with algebra in a way which, had it been understood when 

 these symbols were first used, would probably have always saved them 

 from the distinctive term. 



The roots of equations of the fifth and higher degrees are, properly 

 speaking, transcendental : there is no mode of expression except l>y 

 infinite series. And, generally speaking, and with the exception of a 

 few cases in which modes of expression have been invented and M 

 INVERSE functions are transcendental. And a result of such inversions, 

 even though, from our ignorance of its real properties, it may be 

 expressible by ordinary means, is transcendental so long as that igno- 

 rance lasts. And it is useful to observe that forms of the most 

 different kind may be connected together by such a relation as this, 

 that both are cases contained under the same transcendental. 



To exhibit the arrival of one of these transcendentals of inversion, as 

 they might be called, let us take the equation $x . q>'x = 4>(4>x), where 

 <t>'x means the differential coefficient of ^>.r. A large class of solution.! 

 may be obtained as follows: The equation y logy = chos an infinite 

 number of roots, two at most being real, and all the rest of the form 

 o + /3\/-l. Let a, 6, r, &c., be any of these roots, and let $x bo a 

 function of x formed oa follows : 



where one, two, or any number of roots may be taken at pleasure : ami 

 A, B, &c. are any quantities independent of .r. Let i^-'x be the 

 inverse function of tyr, so that ^(i~'.r) i*.r; (lien ^ (i/>~'x 1) is a 

 solution of the original equation, or fz=ji(<lr- t xl) gives <fx. 4>'x= 

 <t> (4>.r). Now <j/-'x is, when more than one root is used, inexpressible 

 except by infinite series : that in, not merely inexpressible in common 

 algebraical terms, but even with the assistance of logarithms and 

 trigonometrical functions. Nevertheless, as particular cases of this 

 solution, both n.r and VC" **) ara found. 



As science advances, quantities which are now called transcendental 

 will lose the name, and be received among the ordinary modes of 

 expression of analysis. One of the first of these will bo the well-known 

 function of n, which is generally designated by Tn, and is sometimes 

 called the j/nm ma-function, sometimes the factorial function. Its 

 expression ifr~* i.'~ t dx taken from x=0 to x=<o ; and when n is 

 an integer it is simply 2 x 1 x 3 x . , , . x n. But when n is a fraction 



