73S 



PROGRESSION. 



PROHIBITION. 



786 



Thus, the multipliers being 1, 1, J, and 2, the four following sets of 

 terms are in geometrical progression : 



7, 7, 7, 7, &c. 7, 8,y, YJ?,&C. 



3, 1, J, I, &c. 9, 18, 36, 72, &c. 



If a be the first term, and 6 the second, the nth term is a (J-T-a)"" 1 * 

 and the sum of n terms U 



a*-"- (a - 6) or a*-- (b a) 



according as a is greater than or less than b. But when a =4, the 

 sum is of course na. If 6-=-a = r, that is, if the terms be a, ar 3 , ar 3 , 

 &.C., the nth term is or"-', and the sum of n terms is 



1 __ r" r 1 



a - - or o - -, 

 1 r r 1 



according as r is less than or greater than one. If r be less than one, 

 the series of terms, however far it may be carried, never reaches n ~ 

 (1 r), though it may approach this limit with any degree of near- 

 ness by making the number of terms sufficiently great. 

 Thug the following equations, 



though always erroneous, stop where we may, yet can be brought as 

 near to truth as we please by writing down terms enough ozi the 

 second side. In the use of the word INFINITE, as explained in the 

 article on that subject, we may then say that the above equations 

 are absolutely true, if the series be carried ad iiijinitum. The 

 general equation, made absolutely true, after stopping at or* in the 

 series, is 



a ar+' 



= - = a + ar + . . . + ar + : - - 



X ~ 1 T 



Other points connected with this equation will be mentioned in the 

 article SKI:, 



There is no doubt th.it every whole can be subdivided into parts 

 without limit, or, in common language, can be divided into an infinite 

 number of parts. An old fallacy, mentioned in MOTION, receives its 

 explanation from the preceding. If we make a = 1 r, the equation 

 carried ad injmitum becomes 



1 = (1 - r) + (1 r) r + (1 r)r + , &c., ad inf. 



By giving different values to r, we have therefore an infinite num- 

 ber of ways of subdividing unity into an infinite number of parts. If 

 then we take a problem in which an antecedent is followed by a con- 

 sequent ; and if dividing the antecedent into an infinite number of 

 parts, we consider separately the parts of the consequent which 

 belong to those of the antecedent, we shall of course divide the conse- 

 quent into an infinite number of parts. It would be a gross fallacy to 

 infer that the consequent must be infinitely great, because it is pro- 

 duced in a never-ending succession of parts, since that never-ending 

 succession wa produced by dividing the avowedly finite antecedent 

 into an infinite number of parts. No one could fail to detect the 

 following : " Let M be divided into an infinite number of parts, a, 6, 

 c, &c. ; let each of these parts be doubled ; then the result is made up 

 of 2a, 26, 2r, &c., ad infinitum ; consequently 2a + 26 + 2e + , &c., 

 being made up of an infinite number of quantities, is infinite." 

 Nevertheless this fallacy was not only produced in an ingenious form, 

 aa a sophism [MOTION], but has even reappeared in modern times as a 

 serious argument. The sophism is known by the name of " Acliilles 

 and the Tortoise." The swiftest of men runs after the slowest of beasts, 

 without (says the sophism) the possibility of ever overtaking it. For 

 if, when they set out, Achilles be at A and the tortoise at T, then by 

 the time Achilles has run over AT, how fast soever he may run, the 

 tortoise will have gone over some length, say TD ; while the hero goes 

 over TB, his dinner (for dinner he may have out of it, in spite of the 

 sophism) goes over IK', and so on ad iiijinitum. How then, asks the 



III I 



C D E, &c. M 



objector, is it possible that Achilles can ever come up with the 

 tortoise, since it U unquestionable (and this is perfectly correct), that 

 let him go as far as he may, he must always come up to where the 

 tortoise was before he can reach the point at which he is ; so that it 

 requires an infinite number of parts of time (but here the sophism 

 quietly introduces an infinite time) to catch the tortoise 1 The answer 

 is, that Achilles will certainly overtake the tortoise at a finite distance 

 from A, say at M : any contrivance which subdivides A M into an 

 infinite number of parts, does the same with the time in which 

 Achilles runs over A M ; and there is no more reason to say that the 

 time is therefore infinitely great, than to say that AM is made infinitely 

 great by the subdivision. This would be a sufficient answer, since 

 it would throw upon the sophist the onus of showing that the 

 infinite number of parts of time makes an infinite time ; but a more 

 complete answer oonskita in positive proof that it is not so, as 

 follows : 



Let A T '<*> caiied a, and let Achilles move m times as fast as the 



ABTS A*D SCI. DIV. VOL. VI. 



tortoise ; then T B is necessarily the mth part of A T, B c of T B, c D of 

 B c, &c. Hence, if t be the tiuie in which Achilles moves over A T, 

 this time, added to his times of going over I B, B c, c D, &c., or t, t-^-m, 

 t-^-m', &c., make up 



Now if m be greater than 1 (for unless Achilles move faster than 

 the tortoise, it is admitted he can never catch it), the series above 



named is 1 -W 1 -- J, so that the whole time is tm-^-(m l), and 



the whole length AM is am-t-(m l), the same answer as would be 

 produced by common methods. The sophism divides this length into 

 the infiuite number of parts 



a JL JL &o. 



in m 1 wf 1 



and taking the times due to each, 



assumes the sum of the latter to be infiuite. 



In the work of a celebrated political economist there is the follow- 

 ing argument, to show that a tax on wages must fall on the labourers; 

 for if it did not so fall, wages would rise, whence the price of goods 

 would rise, which would again eau.--e a rise of wages, and this again a 

 rise in goods, and so on ad injinitum, which is inferred to be absurd. 

 This is of course precisely a repetition of the preceding case ; and 

 granting all the premises, the conclusion by no means follows. For 

 that conclusion is that the rise would go on without limit, which need 

 not be the case. 



The best way of remembering the summation of a geometrical 

 series is by a verbal rule, as follows : The sum is the difference 

 between the first term in and the first term out, divided by the differ- 

 ence between the common multiplier and unity. Thus the sum of 

 30 + 90 + 270 + 810 is 2430 30 divided by 3 1, or 1200. 



PROHIBITION, a writ to prohibit a court and parties to a cause 

 then depending before it from further proceeding in the cause. It 

 will be convenient to define, 1, out of what courts it issues ; 2, to 

 what courts it may be addressed ; 3, under what circumstances it is 

 grantable ; 4, at whose instance it may be obtained ; 5, at what time 

 it may be obtained ; 6, the form and incidents of the proceeding. 



1. A writ of prohibition usually issues from one of the three 

 superior courts of common law at Westminster. It may issue from 

 the Common -Law Court at Lancaster, and from the common-law side of 

 the Court oT Chancery. 



2. It may be addressed to any other temporal court, such as to the 

 Admiralty Courts, to Courts-Martial, to any inferior court in a city or 

 borough, to the Duchy or County Palatine Courts, to the Stannary 

 Courts, to the Commissioners of Appeals of Excise, to any court by 

 usurpation without lawful authority, or to a court whose authority has 

 expired. When any one has a citation to a court out of the realm, a, 

 prohibition lies to prevent his answering. It seems also that it miiit 

 issue to the Court of Exchequer, and to the Court of Common Pleas ; 

 but not to the Court of Chancery ; nor is there any instance of a pro- 

 hibition to the King's Bench. It may be granted by any of the three 

 superior common-law courts to any spiritual court, and by the 

 common-law courts of Chester and Lancaster to the spiritual courts 

 within the county palatine and duchy. 



3. The writ is grantable in all cases where a court entertains matter 

 not within its jurisdiction; or where, though the matter is within its 

 jurisdiction, it attempts to try by rules other than those recognised by 

 the law of England. Matter may be said to be not within the juris- 

 diction of a court in two senses : 1, when the subject-matter enter- 

 tained is in its nature not cognisable by the court ; 2, where the 

 subject-matter is in its nature cognisable by the court, but lies out of 

 the local district where only that court has jurisdiction ; or, in the 

 case of a court whose jurisdiction is general, when the subject-matter 

 lies in a local district exempt from the general jurisdiction of the 

 court, or where the subject-matter of the cause relates to persons over 

 whom the court has no jurisdiction. 



A prohibition will lie to the courts of Admiralty, if they entertain 

 questions of a contract made or to be executed within the kingdom ; 

 to the county court, if an action be brought in it on a judgment in 

 one of the superior courts ; to a spiritual court, if it takes cognisance 

 of any plea concerning a title to an advowson, or an office, or goods, 

 money, or chattels ; and this applies even in the case of goods or orna- 

 ments given to a church, or matters of a criminal nature punishable 

 only temporarily : in short, as it has been said, anything for which a 

 remedy exists at common law. 



Prohibition lies equally both where the matter of the suit is not 

 cognisable by the court, and where, though the substance is cognisable, 

 matter arises during the progress of it, and is clearly about to be tried, 

 over which the court has no cognisance. Thus, if in a cause properly 

 cognisable by a spiritual court, a question arises, and is necessarily 

 about to be tried, as to the existence of a custom, or a prescription, or 

 the limits of a parish, or where in a suit for tithes there is plea of a 



3E 



