84 



KNOWLEDGE 



[June 30, 



uniform; so that, if we consiJer one instant, the rate at which tho 

 si>ace traversed would change for any given interval of time would 

 bi different from tho corresponding rate at some other instant. 

 Reganiing the matter as illustrating the differential calculus, tho 

 drst thing to be found is a general expression for tho rate of change, 

 — the law determining the space traversed being supposed known. 

 Consider now the following way of dealing with the problem : — 

 At the end of t seconds the body has fallen a space repre- 

 sented bv 



i 9 i\ 

 where p represents the accelerating force of gravity (or numerically, 

 a foot being taken as the unit of length, and a second as the nn'it 

 of time, a = 32-2). A second later the body has fallen altogether 

 a space represented by 



J J (« + 1)', 

 so that m the course of that second the space actually traversed liy 

 the bo<lv is 



i ? (< + 1)' - 1 S t» 



And if daring that second the body moved with uniform velocity, 

 we should at once know what that velocity is. For, when'a body 

 moves uniformly over s units of length in t units of time it moves 

 over - units of length in one unit of time, and - therefore re])re- 

 scnts its velocitv. So that the velocity of our fulling body would be 



if the body had moved uniformly during the second. But this is 

 not the case. The body moves faster and faster as the second of 

 lime is passing ; and its velocity at time t is therefore not obtained 

 by the above process. We should clearly get a better result if we 

 took a shorter interval of time. Suppose we take a very short 

 interval indeed, as a thousandth part of a second. Then we have 

 as before, the space fallen in t seconds 

 - i 3 '•, 

 the space traversed one thousandth of a second later 



and the space traversed in the interval 

 -£-L !/ 



"lOOO * 2 (lOOC)'" 

 So that on the false supposition of uniform velocity daring this 

 minute interval, we get for this velocity, 



Licoo a (iotJO)=J • luoo 



or fff + -^. 



20O0 



This is clearly nearer the truth, because in so short an interval 

 as a thonsandth part of a second the change of velocity is exceed- 

 ingly minute. But still we have not the exact velocity. 



If we had taken a yet smaller interval, as tho millionth part of a 

 second, we should have deduced for the velocity 



gt + - - ^ 



2 (1000000) 

 which is yet nearer the truth. 



And the minnter the interval the minuter tho second fraction 

 becomes, the first remaining unaltered. Also, the minnter tho 

 interval the nearer we get to the trno vnlne. 



Bat there is nothing to prevent us from conceiving that tho 

 internal is taken infinitely minute, in which cacc tho second frac- 

 tion disappears, and also we got infinitely near to tho true value. 

 This valne then is 



9t, 

 and as a matter of fact we know independently that this i« tho 

 velocity acrjaircd by a falling body in the time t. 



Kow the reader will not need to be told that I have not gone 

 through these procOBses merely for the sake of deducing this special 

 result. I want him to convince himself of the reasonableness of 

 the above method, and also T wish him to note that though the 

 reasoning has introduced tho conception of infinitely minute 

 quantilit*, and though the result itself is a Hmiliirj value, yet that 

 re«nlt is none the less eiart. The velocity a body has at tho end of 

 any specified time is real, and not a mere mathematical fiction or 

 approximation. Prepared then to see that a real and exact value 

 can be deduced by a seemingly approximate method, let him 

 consider the following way of treating the very some problem. 



Ix;t » represent the space traversed in time /, <t + .i « the spac« 

 traversed in time ( -^ A ( (where 4 « and .i t are to be looked upon 

 as simple qnantities, which may be read, if wo please, increment of 

 the ipace and inrremtnl of the time; or else, for convenience, 

 ■imply della-tpate and delta-lirne). 



s + As^ig(t + Aty (ii.) 

 and therefore, subtracting (i.) from (ii.), 



A s = ;7 ( A < + I (A ()' 



so that if tho velocity of the body during the interval A t were 



As 

 uniform, this velocity, or — ., would 



= j( + |Af. 



This result, however, will not be true, unless A ( is infinitely 



minute. Let A t bo supposed to be made infinitely minute, in 



wliicli condition csll it dt ; then A s also becomes infinitely minute, 



iiud may be called ds; and we get 



^ = at + l.dt 



dt 2 



= gt, since d t is infinitely minute or nought. 



Now this (luantity — , for which wo have thus obtained a definite 



value (although d s and d t are each evanescent) , is called the 

 differential coefficient of a (tho space traversed) with respect to t 

 (tho time). It is really the rate at which the space is increasing at 

 the time t. 



But tho reader will presently have to consider differential 

 coefficients in a general way. The above illustration has shown 

 him how a differential coefficient is deduced in a special case j and 

 also that a difforontial coefficient, though made up of seemingly 

 evanescent parts, may have an exact value, and (what is yet more 

 to tho purpose) has always a real significance. Tho — of our 



illustration is that real and familiar relation, the velocity of a 

 falling body. And so tho differential coefficients we have to deal 

 with as we proceed, aro real matters, not mathematical fictions. 



But the above case will servo as well to illustrate tho moaning of 

 what is called integration as the meaning of differentiation — tlie 

 process actually followed above. 



{To be Continued.) 



(Bm- WBWt Column* 



By " Five op Clubs." 



IT is remarked by several of our Whist correspondents that reason- 

 ing such as we considered in tho latter portion of our last 

 article, in dealing with play third in hand, second round, is too 

 recondite and elaborate for actual Whist jilay. In reality, however, 

 it only seems i-econdito because considerations which occur as carcl 

 after card falls, aro included in a single discussion, as if thoy had 

 all to be thought over before a particular card was played. As a 

 matter of fact, tho practised Whist player attends to these matters 

 almost unconsciously, making his inferences at tho time, and using 

 them afterwards. Thus Ace being led from Ace, ton, nine, eight, 

 two, by himself, and si.'c, five, three falling from second, third, and 

 fourth ])layor8, he at once notes that either second player or third 

 is signalling, and that foiu'th player is not signalling. Ho feels the 

 absence of the /owr rather than thinks about it. Again, where ho 

 knows that there is no signalling, and a high card is played which 

 does not cover one already played, tho practised Whist player does 

 not require an effort of attention to note that none of tho lower 

 cards are in that [jlaycr's hand— he at once knows it, and therefore 

 acts upon it. 



The following hand illustrates tho way in which inferences are 

 made, and shows howitliey affect the piny. Tlicy seem to require 

 much caro and attention, hut arc all in reality jjci-fcctly simple, and 

 such as the Whist player with sufficient practice will make at once. 



