895 



INFINITE. 



INFLAMMATION. 



866 



tion, to which the mind clings, and of which we do not find the like 

 in connection with the term infinity, we shall, after some further 

 explanation, use the term infinitely small instead of " nothing." 



Our explanation of the term infinite will readily show the meaning 

 of the following assertion ; two infinitely great quantities may have a 

 finite ratio. As follows : when A and B are great their ratio may be 

 nearly, say that of 10 to 7 ; when they are still greater they may be 

 still more nearly in that ratio, and so on ; and their increase may be so 

 regulated that the greater they become the more nearly is their ratio 

 that of 10 to 7 J or as nearly as you please, if they may be as great as 

 we please. Similarly, strictly remembering the preceding conditions 

 for the introduction of the word " nothing," we may allow of the 

 introduction of the following phrase : two nothings may have a finite 

 ratio. This means that A and B, both diminishing together, may 

 diminish in such a way that when both are small their ratio may be 

 nearly, say that of 5 to 3 ; when they are still smaller they may be 

 still more nearly in that ratio, and so on : and their diminution may be 

 so regulated that the smaller they become the more nearly is the ratio 

 that of 5 to 3 ; or as nearly as you please, if they may be as small as 

 we please. 



But the idea of two nothings which have a finite ratio, however 

 strictly defined in accordance with the preceding conditions, shocks 

 even many of those who can grasp the method of using the word 

 " infinity." The absolute nothing of subtraction has possession of the 

 field, and it is not worth while to contest it for the use of a word. The 

 term " infinitely small " therefore supplies the place of " nothing " 

 whenever the latter is introduced under the conditions correlative to 

 the conditions under which the use of infinitely great is allowed. But 

 it must be remembered that if the infinitely small quantity thus intro- 

 duced be added to or subtracted from a finite quantity it makes no. 

 change in the latter ; just as if it were the nothing of arithmetic. A 

 few instances of the development of propositions will now be given. 



1. When A is infinitely small B is infinitely great. As A diminishes 

 B increases, and B can be made as great as you please, if A may be taken 

 as small as we please. 



2. An infinitely small arc of a curve is equal to its chord. The 

 smaller the arc the more nearly are the two in the ratio of 1 to 1 ; and 

 the ratio may be made as nearly as you please that of 1 to 1, if the arc 

 may be taken as small as we please. 



3. Of two infinitely small quantities, one may be infinitely smaller 

 than the other. When two magnitudes, A and B, diminish together, 

 the smaller they are made the greater may be the ratio of A to B, in 

 consequence of B diminishing much faster than A ; and it is possible 

 that A may be made to B in as great a ratio as you please, if both may 

 be made as small as we please. The sine and versed sine of an angle 

 are instances. Both diminish without limit with the angle ; but the 

 smaller the angle the greater the number of times which the sine 

 contains the versed sine ; and this to any extent whatever. 



Infinitely small quantities thus used have been called infinitesimals 

 and a succession of infinitely small quantities, each of which is infinitely 

 smaller than the preceding, is said to be a aeries of infinitesimals ol 

 different orders. Such a series is x, x 1 , .r 5 , Ac., in which, by making x 

 sufficiently small, any one may be made to contain the next as often 

 as we please. The infinitesimal calculus is a name sometimes given to 

 the differential calculus, when presented by means of the theory ol 

 infinitely small quantities, in the manner originally propounded by 

 Leibnitz. 



The preceding considerations refer to the substance of nearly all th 

 disputes which have arisen about the first principles of the differentia 

 calculus [DIFKKRKXTIAL CALCULUS] ; and the different systems noticet 

 in that article (with the exception of that of Lagrange [FUNCTIONS 

 THEORY OF]), spring out of different views of the manner of presenting 

 the same idea. 



In the article ANGLE we have taken notice of the circumstance tha 

 an extension of the word " equal " to infinite spaces which coincide 

 would allow of a proof of the well-known assumption of Euclid 

 [PARALLELS.] Let us suppose two equal angles having their sides 

 infinitely extended. We have then two infinite spaces, of which it ma; 

 readily be proved that either may be made to coincide with the othe 

 throughout ita whole extent. And if any two angles be taken, anc 

 infinite spaces be drawn, it may be easily shown that the infinite space 

 of the greater angle is greater than the infinite space of the less. Th 

 comparison of such infinite spaces is therefore possible consistently 

 witk perfect clearness in the meaning of the terms employed, and 

 simplicity of reasoning which would convince any one who is capable o 

 the most ordinary thought. Had Euclid been accustomed to th 

 modes of thinking which involve the idea of infinite magnitude, under 

 any form whatsoever, it may be reasonably suspected that he woulc 

 have admitted the following axiom, " Magnitudes which can be mad 

 to coincide in all their parts are equal," as applicable to infinite as -we 

 as finite apaces. Not having done so, the adherence to his standar 

 has to this day excluded the only proof of the theory of parallel 

 which does not assume the axiom of Euclid or an equivalent. Fo 

 deinnnstration ace PAR.VI.LKLS. 



There is a word which confusion of ideas is bringing into use in th 

 Hcnae of infinitely small ; namely, homeopathic. The confusion is a, 

 follows. The Hyntem of medicine called hf/mreopathy (which mean 

 treatment by similarity), proceeds upon the doctrine that diseases ca 



ARTS AJCD SCI. 1>1V. VOL. IV. 



e cured by use of the medicines which would produce similar diseases 

 n a healthy person. But the homoeopathic practitioners also hold 

 lat excessively small doses, millionths, billionths, decillionths, of 

 grains, are sufficient for cure. The general public, which more readily 

 pprehends the unusually small amount of the doses than the principle 

 n which they are given, has accordingly appropriated the word homoso- 

 athic as a synonyme of infniitsimal. ' 



INFLAMMATION (from injlammo, to burn). When any part of 

 be body is preternaturally hot, red, swollen, and painful, such a part 

 s said to be inflamed, or iu a state of inflammation ; and when these 

 ymptoms prevail to a certain extent, or affect very sensible parts, that 

 eueral constitutional disturbance called fever is excited. 



Inflammation may be either acute or chronic, circumscribed or dif- 

 used, common or specific. The term common, or healthy inflammation, 

 is applied to all those inflammations which occur in a person other- 

 wise healthy, which run a regular course, are usually of an acute 

 haracter, and terminate in one of the conditions hereafter to be 

 lescribed. Specific, or unhealthy inflammation, unless produced by 

 he direct action of a morbid poison, as that of syphilis, variola, &c., 

 never takes place in a healthy individual, but is always modified by 

 ome pre-existing peculiarity or abnormal condition of the system, 

 requently hereditary, and is generally chronic. Although pain, heat, 

 redness, and swelling, characterise inflammation in its most ordinary 

 orms, it is by no means uniformly attended with all these symptoms ; 

 phis is a circumstance which depends on the anatomical structure of 

 the part affected, and on the duration and kind of the inflammation. 



Termination*. Inflammation is said to terminate in three ways : by 

 resolution, suppuration, and mortification. By the first, which is the 

 most frequent mode of termination, is meant a gradual subsidence of 

 ;he swelling, a diminution of the heat, pain, and redness, and an abate- 

 ment of the fever ; the parta return to their natural size and colour, 

 and no pus or matter is formed. Suppuration is said to have taken 

 jlace when the inflammation goes on to the formation of pus ; the 

 iwelling then becomes more prominent, of a shining red colour, and 

 soft in the centre ; if now no artificial opening be made, the matter 

 obtains exit through one or more orifices produced by the absorption 

 of the walls of the cavity in which it is contained, and the abscess, in 

 popular language, is said to have burst. Mortification is the least fre- 

 juent but most severe mode in which inflammation can terminate, and 

 usually is productive of great constitutional disturbance ; when it is 

 the result of a high degree of inflammation, the attendant pain is ex- 

 ceedingly severe, the bright red colour of the part becomes livid, and 

 vesicles form on its surface ; complete death of the part then takes place, 

 and the pain abates, but the pulse is small and feeble, and great prostra- 

 tion of strength, with troublesome hiccup, are the constant attendants. 



Causes. The remote or exciting causes of inflammation are produced 

 either by mechanical violence or by the action of chemical or other 

 agents ; but it sometimes occurs spontaneously, or without any per- 

 ceptible cause for its production. With regard to the proximate cause, 

 this is a question which is not so easily solved ; it has occupied the 

 attention of pathologist* from the earliest times, and the number of 

 theories on this subject attest the number of those who have interested 

 themselves in the inquiry. The older pathologists imagined that all 

 inflammations were produced by a fluxion, or flow of certain humours 

 to a part, and the peculiar nature of the swelling was supposed to 

 depend upon the kind of humour ; thus blood produced phlegmon, 

 bile produced erysipelas, Ac. After the discovery of the circulation of 

 the blood by our immortal Harvey, Boerhaave appears to have been 

 the first who applied the discovery to the solution of this complicated 

 question ; he supposed that the minute blood-vessels became obstructed 

 by the viscidity of the blood, or where this viscidity did not previously 

 exist he imagined that the larger globules of the blood passed into the 

 small vessels and blocked them up. But change in the consistence of 

 the blood being found inadequate to explain all the phenomena of 

 inflammation, it was supposed that the vessels themselves contributed 

 chiefly to its production, and the doctrine of spasm of the extreme 

 arteries began to prevail. Mr. Hunter considered inflammation to be a 

 restorative principle by which injured or diseased parts are repaired, 

 and the act of inflammation he regarded as an increased action of the 

 vessels, which at first consists simply in an increase or distension beyond 

 then- natural size. 



The application of the microscope to the investigation of the con- 

 dition of an inflamed part has very materially changed the views of 

 pathologiste with regard to the nature of inflammation. If the tissues 

 of an animal in a state of inflammation be examined, the following 

 phenomena may be observed. 1. The capillary vessels of the part are 

 observed to be narrowed, and the blood in consequence flows through 

 them more rapidly. 2. The vessels from being narrower than natural 

 become larger, and now the blood flows through them more slowly 

 but evenly. 3. Changes go on in the capillary vessels, and the flow of 

 blood is observed to be irregular, not even, as hi the second stage. 4. II 

 the observation be continued, the blood is seen to be arrested altogether 

 in the capillaries, and these vessels appear to be distended to the utmost. 

 5. The last and most important phenomenon in the series is, that the 

 liquor sanguinis of the blood is seen to exude through the walls of the 

 blood-vessels, and ig sometimes accompanied with the red blood 

 corpuscles, which pass through either the softened or ruptured walls 

 of the capillary vessels. The most important of these changes, and 



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