545 



DISCONTINUITY. 



DISCOUNT. 



6-10 



right Bur disclaimer was the old form in which the lord took advantage 

 of the forfeiture; but as it was decided that the tenant might be 

 treated as a trespasser, and that notice to quit was not necessary, the 

 more convenient action of ejectment came into use, and the proceeding 

 by writ of right was ultimately abolished. 



One of the pleadings in a suit in Chancery is also called a disclaimer, 

 as where a defendant, in his answer to the complainant's bill, disclaims 

 all interest in the matter in question. [EQUITY.] 



And where an estate is given either by deed or will to a person, he 

 may by deed (which need not be enrolled, or, as it is called, made 

 matter of record) disclaim all interest thereunder. 



An executor is said to disclaim when he renounces probate of the 

 will of his testator ; and where the will contains a devise of lands to 

 the executor, the disclaimer ought to be made by deed to afford 

 evidence, in deducing a title to the lands, of the fact of disclaimer. 



DISCONTINUITY (Algebra, &c.). Continuous changes are those 

 which are so made that no two states exist without every possible in- 

 termediate state having been in existence between them. Thus the 

 square on a line of 4 inches contains 16 square inches, while that on a 

 line of 5 inches contains 25 square inches ; and there is no possible 

 area between 16 and 25 square inches which is not equal to the square 

 described on some line between 4 and 5 inches. That is, if a 

 straight line increase continuously, the square described on it increases 

 continuously. 



The first introduction of discontinuity arises from the attempt to 

 represent all magnitudes by numbers. Arithmetical symbols cannot 

 represent continuous change of magnitude. If a foot be divided into 

 2, 3, 4, &c., equal parts, and so on ad infinitum, there exist infinite 

 numbers of lengths which will not be represented by any whatsoever 

 of the resulting fractions of a foot. Hence the difficulties of INCOM- 

 MENSURABLE magnitudes, which arise from the failure of the attempt 

 to represent flowing or continuous changes by the means of changes 

 which always suppose finite intervals, as in passing from number to 

 number. 



But the arithmetical difficulty, being introduced antecedently to the 

 rxpress consideration of discontinuity, is rarely treated as belonging to 

 this subject. In the higher parts of mathematics the necessity for 

 the consideration of discontinuous expressions began with the inves- 

 tigation of partial differential equations. In the introduction of the 

 arbitrary functions which those equations require, discontinuous 

 functions were thought to be admissible by Euler, an opinion which 

 was controverted by D'Alembert, and supported, conclusively, it has 

 always been thought, by Lagrange, It is our own opinion that not 

 only the arbitrary function of a partial equation, but even the arbitrary 

 constant of a common equation, may be allowed to be discontinuous, 

 unless the contrary be a condition of the problem, expressed or implied. 

 By a discontinuous constant, we mean one which preserves one value 

 between certain limit* of the value of the variable, which then suddenly 

 changes its value, preserving the new value till the variable attains 

 another limit, and so on. 



DISCORD, in music, a sound which, when heard with another, is 

 disagreeable to the ear, unless treated according to the rules of art. 

 Discords are the 2nd, sharp 4th (tritonus), flat 5th (semidiapente), 

 minor or flat 7th, and major or sharp 7th. The ratios of these are 

 9 : 8, 45 :32, 64 : 45, 9 : 5, and 15 : 8. The 9th (9 : 4) is also a discord, 

 and though only the octave to the 2nd, is considered in harmony as a 

 very different interval, and treated in a different manner. The 4th 

 (4 : 3) is either discord or concord, according to the manner in which it 

 is accompanied. [CoscoBD.] Discords commonly, but not always, 

 are prepared, that is, the note which is to become the discord, is first 

 heard as a concord : and their resolution is absolutely necessary ; that 

 is, the discord must pass into a concord, though the resolution is occa- 

 sionally retarded. Examples : 



() (T) (3) 



(8) (2) (3) 



(3) () (3) 



il 7 



The perfect 5th in the chord of f, and the 3rd in the chord of j, are 

 treated, so far as regards resolution, as discords. Examples : 



() (5) (3) 

 -I 



DISCOUNT, a um of money deducted from a debt in consideration 

 of its being paid before the usual or stipulated time. The circum- 

 AIITS A*D SCI. BIT. VOL. III. 



stance on which its fairness is founded is, that the creditor, by re- 

 ceiving his money before it becomes due, has the interest of the money 

 during the interval. Consequently, he should only receive so much as, 

 put out to interest during the period in question, would realise the 

 amount of his debt at the time when it would have become due. For 

 instance, 10(W. is to be paid at the end of three years, what should be 

 paid now, interest being 4 per cent. ? Here it is evident that if we 

 divide the whole debt into 112 (or 100 + 3x4) parts, 100 of these parts 

 will make the other 12 in three years (at simple interest), whence the 

 payment now due is the 112th part of 10,000, or 891. 5s. 9d. The 

 rule is, n being the number of years (or fraction or number and frac- 

 tion), ) the rate per cent., and D the sum due, 



Present value = 



100 D 



~< discount = 



In practice, it is usual not to find the real discount, but to allow 

 interest on the whole debt in the shape of abatement. Thus it would 

 be considered that, in the preceding example, three years' discount 

 upon 100?. at 4 per cent, is 12?., or 88/. would be considered as the 

 present value. 



In transactions which usually proceed on compound interest, as in 

 valuing leases, annuities, &c., the principle of discount is strictly pre- 

 served. The present value in the preceding case is, in its most usual 

 form, 



-, and the discount D ^_? ; 



where p ia the rate per pound (not per cent. : thus it is '04 for 4 per 

 cent.). But recourse ia usually had to the tables of present values 

 which accompany all works on annuities or compound interest. [IN- 

 TEREST.] 



The name of discount is also applied to certain trade allowances 

 upon the nominal prices of goods. In some branches of trade these 

 allowances vary according to the circumstances which affect the mar- 

 kets, and what is called discount is in fact occasioned by fluctuations 

 in prices which it is thought convenient to maintain nominally at 

 unvarying rates. This system is practised in some branches of 

 wholesale haberdashery business, and we have seen a list of prices 

 furnished to his customers by a manufacturer of tools at Sheffield, 

 in which the nominal price of each article was continued the same 

 at which it had stood for many years, while to every different 

 species of tool there was applied a different and a fluctuating rate 

 of discount, this fluctuation constituting in fact a difference of price 

 between one period and another: the rates of discount in this list 

 varied from 5 to 40 per cent, upon the nominal prices of the different 

 articles. 



The term discount is also employed to signify other mercantile 

 allowances, such for example, as the abatement of 12 per cent, made 

 upon the balances which underwriters, or insurers of sea risks, receive 

 at the end of the year from the brokers by whom the insurances 

 have been effected. The word discount is further used, in contra- 

 distinction to premium, to denote the diminution in value of secu- 

 rities which are sold according to a fixed nominal value, or according 

 to the price they may have originally cost. If, for example, a 

 share in a canal company upon which WOl. has been paid is sold in 

 the market for 98/., the value of the share is stated to be at 2 per 

 cent, discount. 



There are one or two old difficulties connected with discount. The 

 first is that of the equation of payments. Were it not for the difficulty, 

 and its principle, this would not be worth notice. It was at one time 

 the custom of the works on arithmetic to point out, when sums of 

 money are due at different times, at what time the total amount is 

 to be paid at once, in such manner that the receiver may gain, 

 by the sums which are prepaid, what he loses by those which are 

 overdue. 



To take a simple case, say it is understood that money makes 

 5 per cent, simple interest, that lOOi is due in three years, and 

 3001. more in seven years. The first rule that was given leads, as 

 the reader knows, to the payment of the whole iOOl. in six years; 

 by which the interest on 300/. paid a year before its time balances 

 that on 100/. paid three years after its time. But this, it is said, 

 was not fair; for not interest, but discount, should be allowed for 

 the sum paid before it is due. That .is, the 400/. paid at the in- 

 termediate time should yield 100/. due'+ interest on it since it was 

 due + a sum which put out to interest will make the 300J. at 

 the end of the seven years. The rule for this case gives a result 

 5-9615574 years, instead of six years ; which will be found to satisfy 

 the conditions. 



In truth, however, it depends entirely upon what the notion of 

 fairness is, whether one rule is better than the other, or whether either 

 will do. And it must be remembered that simple interest is a fiction 

 in real business. A creditor cannot demand more than 'simple interest 

 by law : but it does not therefore follow that because certain money in 

 paid under the name of interest, the receiver will let it lie barren in a 

 bag. But what we say is this : keep to the fiction on which both ml* M 

 are constructed, let all interest-money remain barren, and the two rules 



N N 



