Jd;e 22, 1883.] 



♦ KNOWLEDGE ♦ 



381 



landrail, tlie romance disappears, and pive^ place to a vcrj' 

 different fcelinff. A friend of mine in Lancashire has just sent me 

 his bitter ei])eriences of such a trying ordeal. He is a irreat orni- 

 thological enthusiast, bnt — non mirahiledictxi? — the most unbounded 

 enthusiasm has its limits. It seems that for several nights in suc- 

 cession his slumbers were disturbed by the unceasingand loud com- 

 plaint of a wretched corncrake which planted itself on the grass 

 plot directly under his bedroom window, and kept up a continuous 

 unmusical call throughout the dreary watches of the night. For a 

 time, he bore the affliction with the patience of martyrdom ; bat at 

 length, exasperated beyond all bounds at the audacity and barc- 

 fairilncss of the intruder, he got up and shot it. 



What is the cause of the abundance of the bird this year? I can- 

 not look back upon mauy decades of life myself, but it appears cer- 

 tain that landrails have their periods of tnaxima and minima, as 

 well as sun-spots. Surely the cause is traceable ? Probably Mr. 

 (irant Allen could be per.^uaded to give his opinion on the matter? 

 I can assnro you, Mr. Editor, any remarks of his on the jioint would 

 be most welcome to more than one of your many readers. 



Derby, June 8th, 1883. T. K. DEAtv. 



LETTERS RECEIVED AND SHORT ANSWER.S. 

 W. G. Haskell. Mr. Clodd's valuable papers will be reprinted — 

 probably next fall. — *'Eox. (1) Tiie " one" more fully ac()uainted 

 with the laws and forces of Nature than Newton will, I trust, teach 

 you how to present the higher truth known to you in such a way 

 as to convince Newtonians. (2) Tlie difference between imagining 

 anil conceiving, in such cases as you consider, may be thns pre- 

 sented : — To say Newton imagined that attractive forces act by 

 successive impulses would imply that he thought (or at least thought 

 it possible) that they do so. But he thought nothing so absurd. 

 He conceived them to act in that way, as the only possible way of 

 determining the effect of their action. The result he obtained 

 i--)iilf that conception was in /orce was as incorrect as the conception 

 itself, for it indicated a polygon as the path of the attracted body ; 

 but the curve to which that polygon is shown to approximate is the 

 true path, just as the continuous force is the true limit of the 

 force supposed to act by successive small impulses. — Wilfrip. 

 I believe the double "t" in "Britt" is intended to indi- 

 cate the plural, — though in a rather singular way. — G. B. 

 No doubt a large meteor. — W. You cannot properly denote the 

 force of a blow in terms of weight ; although a particular spring 

 may be just as much deflected by static pressure as by a blow, the 

 same static pressure would deflect another spring of different 

 quality in different degree than it would be deflected by a blow of 

 the same force. .\s to the other questions, they belong to the text- 

 book explanations of definitions, Ac, and cannot conveniently be 

 answered here. — E. M. H. In Humboldt's " Looks into Nature," 

 the nature of Indian edible clay is, I think, explained ; bnt I am 

 away from books just now. — C. F. Hall. Thanks; but at present 

 have barely space for subjects in hand. — W. Hii'slev. The longi- 

 tude is now often determined by lunars. Quite possibly Swedenliorg 

 indicated with more or less distinctness the value of the method. But 

 he can hardlj- have taught Newton, Halley, or Flamsteed anything 

 new on the subject, seeing that one of the special objects for which 

 Greenwich Obseri-atory was founded was the observation of the moon 

 for the express purpose of affording a means of determining the 

 longitude at sea.— E. J. Scott. Sixty oranges at 2 for Id. and CO 

 for 3 for Id., are GO at Id. each and 60 at Jd. each, or 120 at an 

 average of ^jd. each ; 120 at 5 for 2d. are 120 at Jd. ; and ^d. ean- 

 cannot be considered precisely equal to tV''-. S-'' '' ''o^' y°^ ^^''"■ 

 That is the only explanation I can think of, except the statement 

 you ask me to explain, which is in itself sufficient — to wit, 00 at 2 

 for Id. and CO at 3 for Id. come to 4s. 2d., whereas 120 at 5 for 2d. 

 come to -ts. — F. L. Grensted. Letter will appear as soon as 

 possible. — R. E. S. The disadvantage of your optical theory of 

 comets' tails (I suppose 500 have asked my opinion on it) is that it 

 is entirelv inconsistent with fact ; otherwise it is capital. — W. 

 H.S.MoxcK. What ismattcr? Nevermind.— C.T.Davy, T.Weltox, 

 J. Gbeex, M. E., E.Delbeijck, W. Oxlev, C. Nobris, D. Draper, W. 

 Miller, H. Penrith, W. Thomas. Excuse delay in acknowledgment. 

 If you only knew how the correspondence arrears are growing. — • 

 Rob. THoM.'iON. The theory of a primitive impulse on the planets 

 is not now held by any scientific nvan. — 3. B. If a table were made 

 a mile square, and set with its central part in a perfectly horizontal 

 position, and water were poured on to it so as to cover it (allow a 

 rim, please), that water would be two inches deeper at the middle 

 of the table than at the middle of an edge, and four inches deep?r 

 than at the comers.- -A.. KiTSOX. I wrote my " Elementary 

 Astronomy " (Is. 6d.) for the purpose you mention. — C. H. Habdixg. 

 Know of no way but writing to secretary of the E. Institution. It 

 was the plan I followed.^Wsi. Clark. Birds have often been 

 observed to go to roost during total eclipse; bnt it is never anything 

 near as dark as night. 



^ir i»att)ematiral Column* 



GEOMETRICAL PROBLEMS. 



By Rkhati) a. Pbocior. 



Pabt IV. 



WE have hitherto taken theorems involving eract results as 

 our illustrative examples, and we have seen that to such 

 theorems, analytical or synthetical methods, or combinat'ons of 

 both, are applicable with equal advantage. We shall presently 

 discuss other propositions of this sort, and of greater cottplexity. 

 Bnt we must now notice the fact that in certain propositions we 

 have no choice ns to the method of solution. This is almost always 

 the case with theorems involving general results, and with j)roi)!e»ns 

 properly so culled — that is, with propositions in which something is 

 required to be done. Propositions of the former class require the 

 synthetical, propositions of the latter class the analytical method. 

 But, of course, neither of these rules holds, necessarily, in problems 

 of great simplicity, in which only one or two steps separate the 

 data from the quasitum. 



We begin with (in instance of this sort — viz., a very simple 

 theorem in which the relation to be established is general. Suppose 

 we have to prove the following proposition : — 



E.x. 3. — Let A B C, I'tg. 8, be an isosceles triangle, A B leivij 

 equal to A C: produce A B to I), and join D C. Then shall D C be 

 greater than B C. 



There is only one proposition in Euclid which deals with the 

 inequality of two sides of a triangle — viz., prop. 19, Bk. I. It 

 naturally occurs to us, therefore, to apply this proposition. We 

 have to show that D C is greater than B C, and we know from 

 Euc. I., 19, that if D C is greater than B C, then the angle DBG 

 is greater than the angle B D C. Now, our figure shows us D B C 

 as an obtuse angle, and a moment's consideration shows that 

 D B C is necessarily obtuse. For this requires that ABC should 



be necessarilv acute. But the angle A B C is equiil to the angle 

 A C B (Euc. I., 5), and two angles of a triangle being less than 

 two right angles (Euc. I., 17), each of these angles must be less 

 than one right angle. Therefore A B C is acute, and its supple- 

 ment, D B C, is obtuse. B D C is therefore acute, and D C greater 

 than B C. ^.^ ,^ 



We will next try a proposition slightly more difficult. 



j;j_ 4^ l,et P, Fig. 9, be a point which does not lie on either 



diagonal of the quadrilateral A B C D ; then shall the sum of the 

 four lines' A P, B P, C P, and D P be greater than the sum of the 

 'diaqonals AC.BD. 



Here it would serve us nothing to begin analytically by sv,pposing 

 A P, B P, C P, and D P to bo, together, less than A C and B D 

 together. It would be impossible to deduce anything from a general 

 relation of this sort. We must, therefore, proceed synthetically. 



Let be the intersection of the lines A C, B D. 



Then we might first be struck by the fact that D P and A P 

 are together greater than A (J and B together (Euc. I., 21). 

 But then we notice that, on the other hand, C P and P D are 

 together less than C O, O D together. So that unless we can show 

 tin; former excess to be greater than the latter defect, we have 

 proved nothing to our purpose. This method does not seem pro- 

 mising. Nor does it seem likely to be useful to take B P,'P C 

 together and then A P, P D together, comparing these pairs, 

 respectively, with B 0, O C together and A O, O D together. 



Let us try taking alternate lines together, namely, B P, P D, and 

 A P, P C. 'We at once see that B P, P D are together greater than 

 the 'diagonal B D ; and that A P, P C are together greater than 

 A C. Hence, P A, P B, P C, and P D are together greater than 

 A C and B D together. 



(To be continued.) 



