810 



DIET 



DIFFERENCE 



humanity has been evolving, there has been a 

 constant adaptation of taste and desire to the 

 needs of the economy. The natural gustatory 

 inclinations as a rule are a good indication of the 

 bodily wants. As a rule wholesome things have 

 a pleasant taste, and the reverse also holds good. 

 It is all-important, however, that the satisfaction 

 of mere gustatory pleasure be not allowed to mono- 

 polise too much of the energy of any individual. 

 Under these circumstances a surfeit is certain to 

 result. There is a well-known law in physiology 

 to the effect that greater and greater stimuli have 

 to be applied in order to produce a series of equal 

 sensations. It follows that the excesses of the 

 glutton and drunkard are out of all proportion to 

 the actual pleasures these excesses produce, the 

 wise man drinking and eating only in modera- 

 tion. 



See the articles on COOKERY, FOOD, DIGESTION, INDI- 

 GESTION ; also Pavy, Food and Dietetics (1874) ; Sir H. 

 Thompson, Food and Feeding (1880); Sir W. Roberts, 

 Lectures on Dietetics and Dyspepsia, (1885); Fothergill, 

 Manual of Dietetics ( 1886 ) . 



Diet ( Lafe. dies, ' day ' ), a meeting of delegates 

 or of dignitaries, held from day to day, for legisla- 

 tive or ecclesiastical purposes ; the title was after- 

 wards extended to such bodies themselves. The 

 term is applied to the sessions of church assemblies 

 in Scotland, but its chief use is as the specific title 

 of the administrative assemblies of the German 

 empire and some other continental states (see 

 GERMANY). 



Desertion of the Diet, in Scots law. The pro- 

 ceedings under a criminal libel are in Scotland 

 spoken of technically as a diet, and when the libel 

 is abandoned by the public prosecutor, or where he 

 fails to appear, he is said to desert the diet. The 

 effect of a judgment of the court declaring that the 

 diet has been deserted, is to free the accused from 

 prosecution under the particular libel or writ, but 

 not to prevent a new process being raised on the 

 same grounds. 



Dietrich of Bern. See THEODORIC. 



Diez, FRIEDRICH CHRISTIAN, the greatest of 

 Romance philologists, was born at Giessen, 15th 

 March 1794, and educated at Giessen and Gottin- 

 gen, with one short interval in 1813 of campaign- 

 ing as a volunteer. In April 1818 he saw Goethe at 

 Jena, and was directed by the sage to the lectures of 

 Raynouard and the study of the Provencal tongue. 

 From 1822 he lived at Bonn as a prlvat-docent, 

 and in 1830 was there appointed professor of the 

 Romance Languages, and there he died, May 29, 

 1876. His first work, Altspan. Romanzen (1821), 

 was followed by a series of valuable works on the 

 Romance languages, of which the greatest are his 

 Grammatik der Romanischen Sprachen (3 vols. 

 1836-38; 5th ed. 1882), and the Etymologisches 

 Worterbuch der Romanischen Sprachen (2 vols. 

 1853 ; 5th ed. by A. Scheler, 1887 ; Eng. trans. 

 1864). These works discussed these languages for 

 the first time from the comparative historical 

 standpoint, and thus formed a sound foundation 

 for all subsequent Romance philology. See the 

 books on Diez, his life and work, by Sachs (1878), 

 Breymann ( 1878), and Stengel ( 1883-94). 



Difference, CALCULUS OF FINITE DIFFER- 

 ENCES. Difference implies two quantities of the 

 same kind, and means in arithmetic that quantity 

 which must be added to the smaller in order to 

 produce the larger, and in algebra that quantity 

 which must be added to either to produce the other. 

 Thus if the quantities be the numbers 5 and 7, 

 their arithmetical difference is 2, while their 

 algebraical difference may be either + 2 or - 2. 

 The difference - 2 arises from the fact that we 

 may in algebra ask the question what must be 



added to 7 to produce 5? and the answer to this 

 is - 2. 



In certain groups of problems, chiefly relating to 

 series, differences considered in a particular manner 

 are of peculiar importance, constituting in fact a 

 branch of higher algebra, which took its origin in 

 Brook Taylor's Methodus Incrementorum (1715), 

 and is now called the Method of Differences or the 

 Calculus of Finite Differences. This method we 

 shall briefly illustrate. 



Suppose it were required to discover the law of 

 formation, and thence to continue the series of 

 numbers : 



4, 3, 0, 1, 12, 39, 88, 165. 

 It would be wrong to assume that only one law of 

 formation will produce these eight numbers, just 

 as it would be wrong to assume that only one curve 

 could be drawn through eight given points ; but 

 for the full discussion of the difficulty here raised 

 the reader must be referred to the chapter on Inter- 

 polation in any text-book on the subject. We 

 shall, however, show how to find one law of forma- 

 tion, and use our figures to illustrate the elemen- 

 tary notation of the subject. The process is to 

 take the difference between each term and the 

 succeeding one, and so get the first series of differ- 

 ences, or, as it is called, the series of first differences ; 

 the process is repeated on the first differences, and 

 so on, as follows : 



No. of term, 12345678 



Givpn Spi-ips -/ ^1 M 2 U 3 U 4 u s u s ^7 u s 

 ies> \4 3 1 12 39 88 165 



IstDiffprpncp* / AM i AM 2 Aw s AM 4 AM s AM 6 AM 7 

 ,es, < _j _ g j j j* 27 49 77 



2d Differences, j A ^ A ^> A ^ ^ * A ^ ^ 



3d Differences, ( A J* 1 A ^ *% A X *%, 

 ^ o b o b b 



The line of third differences suggests a law of 

 formation, and enables us to continue the series 

 as follows : 



165 276 427 624 

 77 111 151 197 

 28 34 40 46 

 6666 



It could further be shown that if Ux be the zth term 

 of the series above, then 



/// /-y>3 *7^C^ .1. 1 ^y ^ 



The operation indicated by A is defined by the 

 following equation, where Ux means any function 

 of x : 



&Ux = Ux+i - Ux ( 1 ) 



or, if we denote the 1 added to x by AX, by the 

 equation 



or 



a'ux _ 'ux+\ ux , 2 \ 



with the appearance at least of greater gener- 

 ality by 



AMg _ Ux+i - Ux /gv 



As a final example let us suppose Ux = x z - Then 

 we have, using equation ( 1 ) above, 



&Ux ~- \3C ~r~ L ) ~~ 3& 



- 2x + 1 



This is a case of the direct problem of the calculus, 

 but there is also the inverse problem : Of what 

 function is 2x + 1 the difference ? The solution to 

 this is denoted by the symbols : 



or, strictly speaking, for reasons which we need 

 not give, by 



2(2a; + 1) = x 2 + C ; 

 and a; 2 + C is said to be the integrate of 2a; + 1. 



