M7 



AI'PROAt 



APPROXIMATION. 



The main objection* to apprenticeship are, iU interference with the 

 uiuueUjr which erery (nan ha*, or ought to have, in hi* own labour, 

 and ita encroachment not only on the liberty of the workman, but also 

 of thoM who might be disposed to employ him, and who may lately be 

 allowed to judge whether he is fit to be employed or not To require 

 in the more cutnmon mochantoal trade* the same length of apprentice- 

 hip at in the nicer and more difficult art*, U manifestly unnecessary 

 and inexpedient ; and it U obvious that long apprenticeships hare a 

 tendency rather to reprra* than encourage a lore of industry, aa an 

 apprentice U excluded from the greatest incentive to voluntary labour, 

 namely, a participation in the fruit* of hu exertion* or skill. Moat of 

 then objection* apply to apprentioeahip aa a Decenary rather than a 

 voluntary contract, and are of courm removed by the present state of 

 the law. At the ago at which apprentice* are usually bound noun- 

 subjection to restraint is desirable ; and, whether by being bound an 

 an apprentice, or by working as a journeyman, a workman is most 

 lik.-lv to gain a complete knowledge of his trade, and to acquire habits 

 of industry, may be left to the determination of those who are prac- 

 tically interested in the question. 



APPROACHES. This term is applied to the covered roads which 

 are made by a besieging army to protect them from the fire of the 

 place attacked. They consist generally of trenches excavated in the 

 ground, the earth being thrown towards the fortress to screen them 

 from the defenders. Occasionally, as on rocky or marshy ground, they 

 are made by bringing masses of earth to the ground in bags, or by 

 stuffed gabions, wool packs, or any other bulky material which can be 

 easily obtained. For further particulars see SIEGE. 



APPROVER. By our ancient law, where a person indicted for 

 treason or felony confessed the crime charged in the indictment, he 

 might be admitted by the court to reveal on oath the accomplices of 

 his guilt, and he was then called an approver. 



The court might either give judgment and award execution upon 

 him, or admit him to be an approver. In the latter case a coroner was 

 directed to receive and record the particulars of the approver's dis- 

 closure, which was called an appeal, and process was thereupon issued 

 to apprehend and try the appeUea, namely, the persons whom the 

 approver had impeached as the partners of his crime. 



As the approver, in revealing his accomplices, rendered himself liable 

 to the punishment due to the crime which he had confessed, and was 

 only respited at the discretion of the court, it was conceived that an 

 accusation made under such circumstances was entitled to peculiar 

 credit, and the accomplices were therefore put upon their trial without 

 the intervention of a grand jury. 



Here, however, as in other appeals [APPEAL], the accused were 

 allowed to choose the mode of trial, so that the approver might be 

 compelled to fight each of his accomplices in succession. But, unlike 

 an appeal by an innocent person, the prosecution might be defeated by 

 a pardon either to the approver or to the appellee. 



If the approver failed to make good his appeal, judgment of death 

 was given against him. If he succeeded in convicting the appellee, 

 then he was entitled to a small daily allowance from the time of being 

 admitted approver, and to a pardon. 



The appeal by approvers had become obsolete before the abolition of 

 it by parliament ; and the present practice is to prefer a bill of indict- 

 ment against all parties implicated in the charge, and to permit the 

 criminal, who confesses his guilt, to give evidence against his com- 

 panions before the grand jury. If upon the trial the demeanour and 

 testimony of the accomplice is satisfactory to the court, he is recom- 

 mended to the mercy of the crown. (2 Hawk,, Pleat of the CVoirn, 

 ch. 24 ; ' Blackst. Comm.' Mr. Kerr's ed., vol. iv. p. 300.) 



APPROXIMATION, from the Latin, signifies a draining near to. 

 In mathematics, results are said to be found by approximation, when 

 the process employed gives nearly, but not exactly, the result required. 



Strictly speaking, the observed phenomena in every branch of 

 experimental philosophy are approximations, more or less near, to the 

 truth. Thus the distance of the sun, or the diameter of a planet, are 

 only known approximately : but general custom does not sanction the 

 application of the term to any " drawing near " in which the imper- 

 fection arise* from error of the senses, or of instruments. It is only 

 when the defects of mathematical analysis oblige us to be content with 

 a formula which gives result* only nearly true, that the latter are 

 aid to be approximate. To this part of the subject, then, we confine 



It may be stated as a general fact, that there are very few mathe- 

 itical procnssn*, except those of pure geometry, which give absolutely 

 met determination*, in which the answer obtained is neither more 



than is necessary to satisfy the conditions of the question. 

 But the fault is not in the uiavtmim themselves, but in the problem* 

 which it u necessary to submit to them, and in the nature of arith- 

 metical, as distinguished from geometrical, magnitude. It is worth 

 while, briefly, to elucidate thi* point. In geometry, the mind con- 

 ceives one line or angle to differ from another by some niagnr 

 the same kind which can be aatgned, and a magnitude is rather 

 imagined to be given, than actually given. If we attempt to omitrwt 

 the line or angle of geometry, w e must have recourse to approximation , 

 and that of the roughest character, since the errors are as great as 

 those of the sense*. It is only Jiy laying down the postulate that any 

 hne or angle can be assigned independently of all mechanical methods. 



that geometry becomes a science of absolute exactness. In aritl. 

 on the contrary, the very first notion of numbers throws a theoretical 

 difficulty in the way. We can imagine a line to grow or increase 

 ty ; that is, in such a way that it shall not increase from one 



to two feet, without previously assuming every possible length which 

 lies between one and two feet Thi* idea U forced upon us whenever 

 we see points moving to or from each other. But is it therefore true, 

 that every possible length which is greater than one foot and leas than 

 two, can be expressed by one foot and some determinate numerical 

 fraction of a foot f This question reduce* itself to the following. Let 

 AD be 



+ 1 1 1 



A a c 



greater than A B (one foot), and less than A c (two feet) ; if B u be 

 successively divided into two equal parts, three equal parts, four equal 

 parts, and so on, ad injuiitum, does it follow that some one or other of 

 the subdivisions must of necessity fall upon the point D, previously 

 taken at hazard 1 If we appealed to the evidence of the senses, we 

 should certainly answer in the affirmative, for, though the finest 

 compasses were used, we should soon find some point of subdivision so 

 near to D, as not to be distinguishable from it by the severest test our 

 senses could apply. But our mechanical points are minute solids, 

 while, the mathematical point has neither length, breadth, nor thick- 

 ness. Conceive the Utter, and the affirmative answer does not appear 

 self-evident ; for though the continuation of the points of subdivision 

 is unlimited, the number of points which can be taken in the line is 

 also unlimited. But we can demonstrably answer the question in the 

 negative (see the ' Treatise on the Study of Mathematics,' by the Society 

 for the Diffusion of Useful Knowledge, p. 81) : as an instance, let B D 

 be equal to the aide of that square of which B c is the diagonal, or let 

 BOM the circumference of that circle of which B D is the diameter. 

 In neither case can one of the subdivisions of B c ever fall on D. 



Here then U a fruitful source of the necessity of having recourse to 

 approximation, since we cannot be sure that any required relation 

 between concrete magnitudes is absolutely expressible in numbers. 

 In fact, we may state the following as a result of experience, though, 

 not so far as we know, capable of demonstration : numbers being 

 taken at hazard, and submitted to any process which requires the 

 solution of an equation higher than the first degree, the odds are 

 greater than can be assigned against obtaining an absolute result 

 without approximation. In a common table of logarithms, fixing at 

 hazard upon any number, the odds are nearly seventeen thousand to 

 one against choosing a number of which the logarithm can be exactly 

 given. 



This would appear to throw an air of uncertainty over almost all 

 the conclusions of pure mathematics, and justly so, if it were not for 

 the following truth, which, except so far as the labour of approxi- 

 mation is concerned, renders it practically immaterial whether a result 

 is obtained exactly, or by approximation. Any equation whatsoever, 

 which expresses the conditions of a possible problem, if not capable of 

 exact solution, may yet be so far satisfied that a number or fraction 

 can be found, which, on being tried iu the given equation, shall pro- 

 duce an error smaller than any we may think it necessary to name at 

 the outset. For instance, the ratio which the circumference of a circle 

 bears to its diameter does not admit of an exact and absolute deter- 

 mination. If any two numbers be named, their ratio is either too 

 great or too small. But supposing it asked to determine the circum- 

 ference of a circle from its diameter so nearly, that the error shall not 

 be so much as a foot for every hundred miles of diameter, or in that 

 proportion. It can be shown to be more than sufficient for thi.- pin 

 pose to multiply the diameter by 355 and divide by 113; which, if 

 the diameter were 100 miles, would give 314 miles, 280 yards, and 

 1 foot : this, though too small, is within the conditions of the ques- 

 tion, not being too small by one foot. Again, though it is impossible 

 exactly to solve the equation **=? or a? 7 = 0, that is, to find a 

 fraction which, multiplied by itself, shall make 7, yet naming any 

 fraction, however small, at pleasure, for example, one millionth or 

 000001, it is possible so to determine i, that x 1 7, though not abso- 

 lutely Huthiny, shall be less than the proposed fraction, one-millionth. 



It is not our purpose here to enter upon methods of approximation : 

 no space which we could devote to the subject would suffice to explain 

 any of them with sufficient detail to render them of practical use. We 

 shall therefore content ourselves with giving a general view of one of 

 the great methods, we might say, the great method, usually employed, 

 and shall thereby, in succeeding articles, show the young mathematician 

 that various methods, upon which he must have come in the course of 

 his reading, contain a common principle, though disguised under the 

 various forms of calculation which it is necessary to employ in different 

 esses. We must now suppose the reader acquainted with the elements: 

 of the differential calculus. 



\\ h'-n a number is given, and certain processes arc also known, so 

 that they can be performed cither exactly or approximately, we are in 

 possession of the solution of the following question given the number, 

 and the process, to find the result of the process. Hence immediately 

 there results) reason for inquiry into the inverse question knowing 

 the process, and the result of it, what was the number on which the 

 process was employed 1 The way of finding this number is called the 



