INCOMMENSURABLE. 



INCONCINNOUS INTERVALS. 



poor," and (wet 109) " the allotment warden* (appointed by wet. 108) 

 ahall from time to time let the allotment* under their management in 

 garden* not exceeding a quarter of an acre each, to such poor in- 

 habitant* of the pariah for one year, or from year to year, at uch rent* 

 payable at such time* and on such terms and conditions not inconsistent 

 with theproTiaiona of this act, as they shall think fit." Section 112 

 prorides for the application of the rents of allotments; the residue of 

 which, if any, after the payment* mentioned in this section have bean 

 defrayed, is to be paid to the overseers of the poor in aid of the poor- 

 rates of the parish. 



Sections (147, 148) provide for the exchanges of lands not subject to 

 be included under this act, or subject to be inclosed, as to which no 

 proceeding* for an inclosure shall be pending, and tor the division of 

 intermixed lands under the same circumstances. 



Under section 152 commissioners are empowered to remedy defects 

 and omissions in awards under any local act of inclosure, or under the 

 & 7 Will. IV. c. 115 ; and under section 157, the commissioners may 

 confirm awards or agreements made under the supposed authority of 

 A 7 Will. IV. c. 115, if the lands which have been illegally inclosed 

 or apportioned or allotted, shall be within the definition of lands sub- 

 ject to be inclosed under this act. In accordance with this act two 

 commissioners, with a secretary, were appointed, under whose manage- 

 ment its provisions have been extensively carried into operation. The 

 commissioners make yearly reports to parliament of the proceedings 

 which have taken place, and one or more acts are passed every year 

 confirming the inclosures submitted by the commissioners for approval. 

 In their 13th annual report, presented in 1858, the commissioners 

 state that the number of applications of all kinds for inclosures since 

 the passing of the acts had been 2851, and that the whole acreage of 

 inclosures confirmed prior to this report was 226,010 acres, while the 

 acreage of inclosures in progress amounted to 262,418 acres. 



The provisions of this act seem to be well adapted to remedy the 

 evils that were stated in the evidence before the select committee ; and 

 there can be no doubt that agriculture has been greatly improved, 

 the productiveness of the land increased, and employment given to 

 labour by this judicious and important act of legislation. 



INCOMMENSURABLE, INCOMMENSURABLES, THEORY 

 OF. The application of arithmetic to any science of concrete magni- 

 tude supposes a certain magnitude to be taken as unity, and all other 

 magnitudes to be expressed by the number of times or parts of times 

 which they contain this unit. Such an application, therefore, made in 

 the usual manner, requires the assumption of this proposition, that all 

 magnitudes are either fractions or multiples, or compounded of frac- 

 tions and multiples, of any magnitude that may be named. This 

 proposition is not true ; for instance, we shall presently prove that if 

 the side of a square be called 1, no number or fraction whatsoever will 

 exactly represent the diagonal! But we shall also prove that it may be 

 made as nearly true as we please : for instance, that we may find a line 

 as nearly equal to the diagonal as we please, which shall be a definite 

 arithmetical fraction of the side. Quantities which are so related that 

 when one is capable of being represented in terms of a certain unit the 

 other is not, are called incumnunnurablei. The reason is as follows : 

 any two whole numbers or fractions of the same unit must have a 

 common measure ; thus, all whole numbers have the common measure 



1 ; and any two fractions, - and - (a, 6, j>, and q being whole numbers), 



I* 



have the common measure , which is contained exactly aq times in 



the first, and bp times in the second. Conversely, any two magnitudes 

 which have a common measure can be arithmetically represented by 

 the same unit : for if A and B have the common measure M, and if this 

 measure be contained 7 times in A and 10 times in B, then it is evident 

 that by taking M as the unit, A is represented by 7 and B by 10. If, 

 then, there be two magnitudes which cannot be represented by mean* 

 of the same unit, they cannot have any common measure whatsoever, 

 and are therefore inr,,mmrniuraf,le. It also follows from the preceding, 

 that any twu commensurable magnitudes must be to one another in 

 the proportion of some one whole number to some other whole 

 number. 



To prove that there are such things as incommensurable magnitudes, 

 we shall take the 117th (and last) proposition nf the tenth book of 

 Euclid, which demonstrates that the diagonal and the side of a square 

 are incommensurable. Let t> be the diagonal and s the side, and if 

 they bo not incommensurable let a and x be the whole numbers to 

 which they are proportional ; that is, let M be a common measure, and 

 let D and severally contain M, a and x time*. Then the square on D 

 will contain the square on M no time*; and the square on 8 will contain 

 the square on M xx time*. But the square on D is double of the square 

 on ; thereforu aa is twice .rr. Now, let a and x have no whole 

 common measure except unity, which may be supposed, for if they 

 have a common measure, we may divide both by it, which will give 

 two whole number* in the same proportion, and so on until no common 

 measure i* left. Tli.-n. because a times a is double of .r times x, a 

 time* a i* in even number ; whence a is an even number, for if a were 

 odd, a time* a would be odd. Therefore, x is not an even number, 

 fur if it were, a and x would have the common measure 2 ; whence x 

 is an odd number. Let i be the half of a, which is a whole number, 



ince a is even ; whence a = 2*, and aa = 4 M, which is aUo 2xr , and 

 thence it follows that xx= 2M. Therefore, xx is an even number, and 

 x also ; for if x were odd, x* would be odd : whence x is even. But 

 it was just now proved to be odd ; so that the same number is both 

 odd and even, which i* absurd. Thi* contradiction follows whenever 

 we suppose s and D to be in the proportion of any two whole numbers; 

 consequently, 8 and D are not in the proportion of any two whole 

 numbers, and therefore are incommensurable, for if they were com- 

 mensurable they would be in the proportion of some two whole 

 numbers. 



We have next to prove that any two magnitude* whatsoever, being 

 incommensurable, may be made commensurable by as small an alteration 



Kl 1 



O L 



A! ! 1 1 1 1 i 1 1 i i ! 1 1 



as we please in either. Let A and B be two incommensurable magni- 

 tudes, and let K be a third magnitude of the same kind, which may be 

 as small as you please, provided only that it be given and known. 

 MTI .] Now, some aliquot part of A must be less than K ; if 

 not the hundredth, try the thousandth; if not the thousandth, try the 

 millionth, and so on. Whatever K may be, it is possible to divide A 

 into equal parts, each of which shall be less than K. Let M be such an 

 aliquot part of A, and having divided A into its parts, set off part* 

 equal to M along B. Then A and B being incommensurable, B will not 

 contain M, the measure of A, an exact number of times, but will Ho 

 between two multiples of M , say B o and B L. From this it is obvious 

 that B does not differ from either BO or BL by so much aa c i.. .mil 

 therefore not by so much as K. But B o and B L are both commen- 

 surable with A, since all three are multiples of M. Here, then, are B a 

 and B L, the first a little less than B, and the second a little greater, 

 neither differing from B by so much as K, but both commensurable 

 with A. Thus it is also evident that two whole numbers may be found 

 which shall be as nearly as we please in the same ratio as two given 

 incommensurable quantities. 



The difficulty thus inherent in the application of arithmetic to 

 concrete magnitude is not met with in practice, because no case can 

 arise in which it is necessary to retain a magnitude so closely that no 

 alteration, however small, can be permitted. But in exact reasoning, 

 where any error, however small, is to be avoided, it is obvious that the 

 arithmetic of commensurable magnitudes, and the arithmetic (if there 

 be such a thing) of inco mmensurable magnitudes, must not be con- 

 founded. The difficult/ was overcome by Euclid, in the manner 

 pointed out in the arti jlo PROPORTION, so completely and effectually 

 that nothing has been a Ided to his solution of it except unsuccessful 

 attempts to evade it. 1 hose who avoid the fifth book of Euclid gene- 

 rally substitute either the tacit assumption that all magnitudes are 

 commensurable, which is not true, or some play upon words, which a 

 person who feels the rig our of Euclid places on the same shelf with 

 nature's horror of a vac mm and other explanations of the same kind. 

 We could even point out a celebrated work on geometry which 

 expressly rests on being able to make its errors too small to be per- 

 ceived by the senses, and asks for no other reception of propositions 

 which involve incommensurables. 



The doctrine of incommensurable quantities was carried by Euclid 

 to an extent which would excite as much admiration as any portion of 

 his writings, if the tenth book were generally known and read as 

 the production of a person unassisted by algebra. [IIUIATIONAL 

 QUANTITIES.] 



INCOMl'ATIBLES, in Materia Medica, applied to those article 

 which are considered improper to be united in the same prescript ion. 

 Strictly speaking, the term applies only to introducing into the same 

 formula articles which exert a chemical action on each other, and so 

 produce a result or compound of a useless or hurtful kind. Two or 

 more articles may be introduced into a prescription, which by their 

 combination neutralise the properties of each other. The resulting 

 compound may be perfectly insoluble in the juices of the stomach, so 

 as to be inert or hurtful by its insolubility. But this result is often 

 sought on purpose to neutralise acrid or corrosive substances ; such as 

 when chalk or lime from a wall is given to a person who has swallowed 

 oxalic acid. [ANTIDOTES.] But entire loss of power does not invariably 

 result from combining substances which chemists deem incompatible ; 

 thus chalk and opium form a more powerful astringent than 

 singly given ; and opium and acetate of lead in warm water form a 

 fomentation of much use in erysipelas. But that utterly inert com- 

 pounds often resulted from bringing together numerous ingredients is 

 certain, as seen in the Polypharmacy of the ancients, of which the 

 famous Mithridate is an instance. Even this has been surpassed in 

 modern times, some prescriptions of Huxham containing above four 

 hundred ingredients. On the opposite hand, excessive simplicity is 

 perhaps too much aimed at in the present day. Dr. Paris' ' Pharma- 

 cologia' may be advantageously referred to; and Translation of the 

 ' London Pharmacopoeia,' by Richard Philips. 



1NCONC1NNOUS INTERVALS, in Music, are sounds which agree 



