S65 



TRIGENIC ACID. 



TRIGONOMETRY. 



or moulded piers between them. In English churches triforia are, 

 during the First Pointed period, very ornamental in character ; in the 

 Second and Third Pointed styles they are generally smaller and less 

 important. 



TRIGENIC ACID (C.H^OJ. Aldehyde absorbs the vapour of 

 cyanic acid with great avidity, producing a white crystalline mass, 

 whilst carbonic acid is disengaged. Amongst other substances, trigenic 

 acid is formed by the following reaction : 



0,11,0, + SCyHO., = C 8 H,N.,O 4 + 2CO a 



Aldehyde. Cynnic acid. Trigenic acid. 



Trigenic acid crystallises in small prisms, very slightly soluble in 

 water or alcohol. When submitted to destructive distillation, it 

 appears to yield quinoleine and cyanuric acid. The trigenate of silver 

 is insoluble in cold water, but soluble in hot water. 



TRIGLYPH. [GRECIAN ARCHITECTURE.] 



TRIGONOMETRICAL CO-ORDINATES. These co-ordinates were 

 invented, independently of each other, by Professor Gudermann, of 

 Cleves, and the Rev. Charles Graves, of Trinity College, Dublin. The 

 latter called them spherical co-ordinates, a term which is liable to be 

 confounded with the common astronomical co-ordinates described in 

 SPHEBE. This remarkable extension leads to a system of algebraic 

 geometry for curves on a sphere, singularly resembling in its results 

 the common system in a plane, many of the formula) of the two being 

 absolutely identical We venture to predict that many difficulties of 

 common algebra, arising from the entrance of the consideration of 

 infinity, will find easy and natural explanations when the common 

 system is considered to be, as it really is, an extreme particular case of 

 this more general view. 



Through any point o on a sphere, let arcs o x and o r be drawn, 

 which represent the axes of x and y. Let o X and o r be each of them 

 quadrants, and through any point p let arcs Y M, X y be drawn. Then 

 the trigonometrical co-ordinates of the point P are the tangcnU of o M 

 and o N, or of the angles subtended by them at the centre. These 

 tangents are called x and y. If the whole system be projected, by 

 lines drawn from the centre of the sphere, upon the tangent plane at 

 o, then on and ox will be projected into rectilinear co-ordinates to the 

 projection of p. The equation of a great circle is of the first degree, 

 or of the form ax + by + c = 0, and an equation of the nth degree be- 

 longs to the intersection of the sphere with a cone of the nth degree 

 whose vertex is at the centre. In the polar co-ordinates o p La the 

 radius of the point, and the angle p o M its angle This system of polar 

 co-ordinates had been previously considered by Mr. T. S. Davics 

 (' Trans. R. S. Edinb.,' vol. xii.), at great length, and with valuable 

 results. Mr. Graves published his account of the trigonometrical co- 

 ordinates in the appendix to his ' Two Geometrical Memoirs on the 

 General Properties of Cones of the Second Degree,' Ac., Dublin, 1841. 



Thu complete algebra [NEGATIVE, Ac., QPANTITIES] may be easily 

 applied to this system. If, in fact, the tanyent of o P were culled the 

 radius vector, instead of o P, and denoted by r, the angle POM being 

 9, we should have, for rectangular co-ordinates, exactly us in the plane 

 system, 



x = r COB 9, y s= r sin 6, x+ y V-l =sr' v '- 1 



TRIGONOMETRICAL CURVES, a name given to curves having 

 such equation as y = sin x, y=coax, y = a cos x + b cos 2x, Ac. To 

 construct the forms of such curves from the knowledge of the funda- 

 mental properties of the sine, cosine, Ac., should be an early exercise 

 of the student in trigonometry, and will be of use in his subsequent 

 reading. 



TRIGONOMETRICAL SERIES. Infinite series which are of the 

 form a sin x + b sin 2jc + c sin 3.C + Ac., and a cos x + b cos 2.e + c 

 cos Sx + 4c., are of important use in the higher parts of mathematics. 

 The common mode of finding their equivalents, when a + bz + c + Ac., 

 admit* of representation in a finite form, is very easy. For instance 

 let it be required to find 1 + a COB x + o* cos 1x + , Ac. Assume 

 2 cos x = z + z-', then 2 cos nx = *" + *-", and substitution gives for 

 the whole series, 



1 1 2-0 ( -i-z-i) 



which is 



r 2(1-01 



1 O COS X 



+ a*) 



1 2 a cos x -f a" 

 The most remarkable property of these series a that they are capable 



of representing discontinuous lines, so that an arc composed of arcs of 

 different curves might have every one of its points made to satisfy an 

 equation of either of the preceding forms. The whole of the discon- 

 tinuous undulation, for instance, drawn in ACOUSTICS might be in- 

 cluded under one equation. See the ' Differential Calculus ' (' Library 



Universal Knowledge '), p. 621. In the higher parts of physics, this 

 property is of the greatest importance ; and without doubt it is one of 

 the most remarkable in the whole range of analysis. But it will not 

 perhaps appear so singular if we remember that every curve made of 

 arcs of different curves can have a continuous curve, represented even 

 by a common algebraical equation, drawn as near as we please to any 

 collection of its arcs. Lagrange showed the use of a finite tri"ono- 

 metncal series to be a very easy mode of actually representing the 

 ordmate of this approximate curve : the infinite trigonometrical series 

 i the limit which actually attains, algebraically speaking, the perfect 

 representation of that to which a finite number of terms is only au 

 approximation. 



TRIGONOMETRICAL SURVEY. [GEODESY 1 



TRIGONOMETRICAL TABLES. The chronological list given in 

 TABLES will serve as a sketch of the history of these tables : we desire 

 here to elucidate a point of their construction which, in the present 

 state of transition from one system of definitions to another [Taioo- 

 NOMETRY], causes a great deal of confusion. 



In the ordinary trigonometrical tables is set down the common or 



iggs logarithm of the sine, cosine, Ac., of every angle which is an 

 ;xact number of minutes (or seconds, or ten seconds, as the case may 

 be) from to 90. Looking into a table we find for the logarithm of 

 !5 P , for instance, 97585913. This number is the logarithm 

 of 5/35/64363, a number containing nearly six thousand millions of 

 units. But the constructors of these tables used a radius of ten 

 thousand millions of units, and their assertion consequently is, that if 

 a right-angled triangle have ten thousand millions of units in its hypo- 

 thenuse, and 35 degrees in one of its angles, the side opposite that 

 angle will be 5735764363, which is correct within one unit. The 

 logarithm of this radius is ten, and the earliest tables were constructed 

 B to give ten figures of the logarithms, from -which ten significant 

 igures of the sine, Ac., might always be found : this radius was there- 

 fore convenient. Those who use the old system strictly, and employ 

 the radius in every formula, find no difficulty : thus, if c be the hypo- 

 thenuse of a spherical triangle, and a and b its sides, we have 



cos c 



cos a . cos b 



log cos c = log cos n + log cos b log (or 10). 



But those who use the old system, and have also dropped into the 

 habit of making the radius always unity, or omitting it from the 

 formula-, and those who use the new system, in which the sines Ac 

 are numerical representations of ratios, will always find a difficulty 

 until they establish a new explanation of the characteristic of Wa- 

 nthms which they find in their tables. We speak of course of be- 

 ginners, for practice will get over such a discrepancy, or will perhaps 

 cause a sufficient explanation to suggest itself. 



,?? ei cr of the two last-mentioned classsg of persons, the sine of 

 6730784363 and its real logarithm is the negative quantity 

 75359 13-1 or--2414087. To them therefore the simple explana- 

 m of the discrepancy between their logarithm and that of the tables 

 The tabular logarithm is always 10 more than the real 

 ithrn, and the real logarithm always 10 less than the tabular 

 ogarithm. There are two ways of proceeding : either to take out the 

 real logarithm, which can always be done, using the characteristic -1 

 (or for distinction, 1) instead of 9, 8 or 8 instead of 2, -3 or Fin- 

 stead of 7, and so on ; or to remember that each logarithm is 10 too 

 great, and to make the correction, either mentally or at the end of 

 each logarithm. We have always found the first mode the better of 

 the two, and we should recommend no one to reject it without a 

 sufficient trial. 



For example, suppose it is required (,,. calculate tan fl=V(sin 1 

 sin 14 H- an 3) 3 . We have (not using the arithmetical complement) 



log sin 1" 2-2418553 

 log sin 14 1-3836762 



add is-6255305 

 log sin 3 27188002 



subtract 2-9067303 



0=1 19' 2)4-7201909 

 2-3600955 



In looking for the result, we remember that the tabular logarithm 

 answering to 2-3600965 is 8'3600955. 



Fur the multiplication and division of negative characteristics, see 

 LOGARITHMS, USE OP (col. 336). 



TRIGONOMETRY. This word signifies the measurement of tri- 



