442 



KNOWLEDGE 



[Dec. 1, 1882. 



*ec«l-l«g. The present statuette appears to have boon inscribed for 

 line .l"ie-ni<-(, •• justilied." Some of the hieroglyphs are iinintelli- 

 f»ible in the copy, and the inscription is e\-idently not complete. 

 There is some mention of a punegyry, or religious festi\-al, but I 

 c-annot make out the connection. 



There is nothing very curious about the statuette in question. 

 Each deceased person had several such, in various materials, buried 

 with him in his grave ; and there arc many hundreds of them in 

 overy Egyptian Museum. 



The dogma as to the fields of Aahlu was borrowed by the Greeks 

 at a late period, and by them converted into i^hjsium. A. B. E. 



BOTANY. 

 Alliiu (021) is correct in supposing that the carbonic acid 

 fipintl by plants is due to decomposition of organic matter. In 

 all seedlings, and in the various parts of the i)lant, such as the 

 llower and fruit, which are incapable of themselves of absorbing 

 their own nourishment from the air, the presence of oxygen is 

 necessary to convert the material in the other portions into the 

 tissue of these growing parts. Carbonic acid is e.\pired at all 

 times, but is only inspired under the influence of light, bat 

 this inspiration is greatly in excess of the expiration of the 

 Bame compound, and it is quite possible that the carbonic acid 

 expired by one class of cells may simultaneously be absorbed by 

 another and contiguous class. It appears, then, that the breathing 

 of plants is not quite analogous to that of animals, for while 

 in the latter the inspiration of oxygen is accompanied by destruc- 

 tion of matter, and subse<|uent removal as carbonic acid, in plants 

 we have a conversion of matter into other forms which are for 

 the most part retained, a small portion only being wasted as gas. 

 Furthermore, we find a double action occurring in plants— absorption 

 and expiration of oxygen and of carbonic acid, which result iu the 

 production of new tissue, whereas in animals we have only iuspiru- 

 lion of oxygen and expiration of carbonic acid, and the accom- 

 panying changes are destructive. The distinction drawn between 

 ]>lants and animals was not intended to be absolute, as will be seen 

 on reference to the last sentence, which commences with "the great 

 iliffcrence; " the insertion of any sentence which could be construed 

 to mean that there was no resemblance between plants and animals 

 being, as it was thought, specially avoided, as I was well aware of 

 the exjiiration of carbonic acid (indicative of oxidation) by the 

 leaves, as mentioned in one of the preceding paragraphs ; as also of 

 the evolution of the same gas by the root.s, whereby the preparation 

 of plant food in the soil is forwarded, but concerning which no 

 was made, as the article was already sufficiently long. 



E. W. P. 



<!Pur iHatftnnatiral £oIumn. 



EASY LESSOXS IN THE DIFFERENTIAL CALCULUS. 



No. XIII. 



"^T EXT take the following illostration of a differential calculu 

 J.1 which is in many respects more important still than that coi 

 •tdered in oar last. 



YI 



Fig. 1. 



latti-vl lit takin;r for the d'-pcndcnt variable the ordinate of a 

 irrc, lake the area aa aAP M (Fig. 1) ninaiiured from some fixed 

 -'linati,' A a to the ordinate P M corrcufMinding to varying value of 

 ■e abaciM O It or x. Potting M N - Ax, we have, if 



u = arcaAPSIa 



1+ Au = area AQ Na 



A» = areaMPQN 



Au nroaMPgN 



Ar MN 



side n N of a rect. »ii N 



equal to Si PQN. 



Now manifestly the smaller JI N is taken the nearer is the rect- 

 angle P N in area to Jl 1' Q L : it is not merely that the difference, 

 the area P Q L (P Q curved) is absolutely less, but it manifestly 

 bears a constantly diminishing ratio to the area PN, until finally, 

 when il N is taken small enough this ratio will bo a vanishing one. 

 Uence while 



— - = n N, we manifestly have — = P M = y , 



A* ■ dx 



or the differential coefficient of the area between a curve, the axis 

 of .r, a fixed ordinate, and the vai-iublo ordinate is this last-named 

 ordinate. Here the language and notation of differentials are, 



I think, simpler and more natural than those belonging to tho 

 method of coefficients. We should say simply du the differential 

 of tho area, that is tho small increment of tlio area, is equal to 

 1/ dx, that is to the area under y and dx the differential of ai, which 

 is the same as saying that ultimately area P Q N M = rectangle P N. 



But now notice tho power wo have obtained for determining not 

 only areas of surfaces bounded by curved lines, but also any function 

 which might be symbolised by such areas. When wo liavo written 

 of any such area as APM«, or of any function which may bo 

 represented by such an area, the relation 



'£-" 



where y is some function of x, we can tell at once what « is, if only 

 wp can determine what function has y for its differential coefficient. 

 y,„. ,..,. 1.-., .. ......Ti tliat no two quantities can have the same differ- 



• unless either they are equal or differ only by a 

 r trically this is much tho same as saying that 



II .1 A PM a has PM the ordinate of curve APQ for 

 I Ik iu(:i:iiin' (ji its rate of increase, except some area, as OBPM, 

 E U I' M, or tho like, differing from a A P M by some quantity 

 which remains constant while wo vary M, and with it P M.) 



Now although determining what quantity that is which has 

 some given function for its differential coefficient is by no means so 

 easy or so sore a process as differentiating, yet in a great number 

 of cases integration can be readily managed ; in others it can bo 

 accomplished with more or less difficulty, and in yet others, 

 approximations can bo made to the desired integration, and the 

 result we require can be determined with all necessary approach to 

 exactitude. 



The notation for the converse process to differentiation, or inte- 

 gration is as follows : — When wo have a function u such that 



! express the same relation by saying that 

 ' dx. 



To bo more precise wo may ■ 





i 



dx + a. constant, 



but the constant is in reality understood 



prcssi 



We mav find a geometrical illustration of thi 

 = Ajdx 



./, 



the other form of ex 

 |uation 



.V I'Q 1)0 a curve as before, 0M = 3;, VM = y, M»= Am. 

 A Q N a is the sum of a number of small rectangloa like 



Pn, when these are made indefinitely thin. Now 



I 



