July 13, 1883.] 



♦ KNO\VLEDGE ♦ 



31 



all) seem to have collected everything written by men of their race, 

 as of superhuman excellence, and many outside the race accept 

 that view ; others do not : with the farmer such matters as your 

 father dealt with are very properly regarded as " of faith " ; with 

 others they are not so regarded at all. These last take little interest 

 in inquiries of the sort. It seems to them the merest accident that 

 instead of them, such matters as the proper interpretation of the 

 Homeric records have not come to be questions of faith. We must 

 Consider these, who are many — even among those who are supposed 

 to view matters otherwise. — A. Andrews. I am sorry, but it is con- 

 trary to " our " rules to answer queries of the sort. — H. Askew, 

 Any proof by which it is shown that the sum of the roots of an 

 equation of the ?ith degree is equal to minus the co-efficient of x"~', 

 will serve to show that the sum of the nth roots of unity, and the 

 sum of their reciprocals, are each equal to zero. — H. C. Jones. 

 Your society for correcting, detecting, and, where necessary, expos- 

 ing slander, has my heartiest good wishes ; but my time is so fully 

 occupied that my joining it would be useless. Many thanks for your 

 kind words and good wishes. — W. T. Southwakd. "Most men" 

 and " a large majority of your fellow-countrymen ! " We question 

 each point ; but the two statements are very different. Albeit 

 Knowledge is meant for those who take an interest in science. 

 "Most men" and "the large majority of our fellow-countrymen ' 

 do not take the least interest in science. More are students of 

 science now than were so twenty years ago ; but they are not one 

 in a hundred of the population yet. Can you imagine that in 

 catering for these, I am likely to consider those others, even if 

 most of them do hold, as you say they do, the ideas you mention P 

 The only way in which those ideas can ever become matter of 

 scientific inquiry is in the way in which they have been touched on 

 here. Their origin, the way in which they have spread, have 

 become modified, have died out among those who inquire, and so 

 forth — these are matters of scientific interest, because, among 

 the subjects of inquiry which science takes up, all relating to 

 man and his ways must be included. If you regard as 

 sneering every expression implying views which do not hap- 

 pen to agree with your particular views, it is unfortunate ; 

 but the fault is not here. Of myself personally I may say (so far 

 as I can judge myself), what I recognise as true of Messrs. Slack, 

 Williams, Clodd, Grant Allen, Wilson, and others whose personaliiy 

 comes most clearly before the readers of Knowledge, that we are 

 one and all exceptionally free from any concern or trouble of mind 

 because others may not view matters precisely as we do. For 

 instance, it appears you differ from me in some points on which 

 you have touched. Very well. What can it matter ? If what I 

 say does not commend itself to you, it does «of, and there an end. 

 No sneer whatever is intended when I say that, while preferring 

 my own views, I have no wish to controvert yours. — H. Malim. 

 No : tiie velocity in the original horizontal direction is affected by 

 gravity, which oidy acts at right angles to that part of the motion 

 at the beginning. See analytical solution. — E. P. Toy. I am 

 much obliged to you ; but travel about so much that fear the 

 book might not be returned in due time. May I be permitted 

 to mention here that your letter gives strong avouchment 

 of tho truth of the account of Mrs. Croad's thought-reading 

 powers ? — P. Pekcival. The statement has never, I believe, been 

 challenged. It was discussed a few years ago before tho Astrono- 

 mical Society, but not questioned. Dr. Warren de la Rue brought 

 it, if I remember rightly, before the notice of tho society. Hut while, 

 so far as I know, no one doubts or denies the occasional existence of 

 organic matter in meteorites, what has been challenged and tho- 

 roughly controverted is the statement made a year or two ago, that 

 the remains of lower types of vegetable and even animal life had 

 been detected in meteorites. — C. .1. B. Your question cannot be 

 answered ; tho velocity of the foot varies from through a maximum 

 to 0, the average rate being 2 yds. (the actual di.stanco covered by 

 tho moving foot in a so-called one-yard step) in 41- I5ths of a second, 

 the time during which one foot is moving and the otlier at rest. — 

 A Weak One. So hard to get space ; but soon. — G. M. Mr. Clodd's 

 " Jesus of Nazareth " is published by Messrs. Kegan Paul & Co. — 

 G. G. Hakdinoiiam. It would not be very easy to answer your 

 (juestions about star-drift in convenient space here. Maps showing 

 tlio star-drift for all the stars whoso proper motion has been deter- 

 mined appear in my " Universe of the Stars." 



Sun-EDITOBIAl. 



J. S. W.— A. F. 0.— J. Habbington.— M. N. E.— A Blind One.— 

 T. .1. R. — Perple.ked. — P. Mackay. The Editor ia not a medical 

 man, nor am I. 



Errata. — In the solution at pp. 10, 11, the following errata 

 occur: P. 10, line 19 from bottom, for \hc.ahC, read \hc.hC ; 

 p. 11, lino 11, for "twice" read "half." 



®\\x ifiattjematical Column. 



GEOMETEICAL PROBLEMS. 



By Eichakd A. Peoctok. 



Part VII. 



THE result obtained at p. 15, fitly introduces us to an important 

 class of problems — viz., those in which we have to show that 

 certain lines, areas, &c., are the greatest or least which can be con- 

 structed under certain assigned conditions. There are few problems 

 of this sort in Euclid. In fact, the seventh and eighth propositions 

 of the third book are the only theorems in Euclid expressly dealing 

 with geometrical maxima and minima. But many interesting de- 

 ductions involve such relations as we are speaking of, and it is well 

 for the student to know how to deal with them. 



It will be noticed that some of the problems already dealt with 

 may be presented as examples of geometrical maxima and minima. 

 For instance, Ex. 4 may be presented in the following form : — 



Ex. 9. — From a point ivithin a quadrilateral lines aredrav.-n to the 

 angles of the quadrilateral ; shou' that the sum of these lines will be 

 a minimttm ivhen the point is at the intersection of the diagonals. 



Presented in this form the problem would be solved precisely as 

 Ex. 4. But suppose it had been given in the following form : — 



Determine a point mithin a quadrilateral such that the sum of the 

 lines from the point to the angles of the quadrilateral shall he a 

 ininimuni. 



Here assuming the student to have no knowledge of the property 

 established in Ex. 4, the problem is not quite so simple. Let us 

 see how it is to be dealt with. 



Fig. 14. 



Draw first the quadrilateral, A B C D (Fig. 14), and from sonio 

 assumed point, P, draw P A, PB, P C, and P D. Then we have to 

 inquire how to shift P so as to lessen the sum of the distances, 

 PA, PB, PC, PD. 



A very short inquiry suffices to show that we shall not gain much 

 information by considering the lines PA, P B, PC, and PD in 

 adjacent pairs. The inquiry might run somewhat in this way : — If 

 P be brought towards B A, the sum of the Unes P B, P A will 

 diminish ; but the sum of the lines P C and P D will increase. We 

 have no obvious signs showing whether the diminution or increase 

 be the greater. Therefore we are not tempted to continue this 

 mode of inquiry. 



Can we, then, by taking the lines in alternate pairs, diminish the 

 sum of one pair without increasing tho sum of the other ? By 

 bringing P towards tho line U 1) (which we draw, at tliis point of 

 the inquiry), tho sum of the lines B P, P D, is diminished (Euc. I., 

 21). Now if this were done without any attention to the lines PC, 

 P A — for instance, if P moved to Q — it would not be easy to assert 

 that the sum of the four distances from the angles was diminishing. 

 But if P be made to move along PA, as to P,, then — since C P is 

 less than P C and P P, together — we are diminishing, not merely 

 the sum of the distances from B and D, but the sum of those frrm 

 and A. So long, then, as wo continue this process, we cannot be 

 going wrong. So that if we bring P to P.; — the intersection of 

 P A and B D. — we have diminished the sum of the distances as 

 much as t/u.i process allows us to do. It is now obvious that by 

 shifting our point from P; towaids A C, along the line 

 P.. B, we are yet farther diminishing tho sum of the ilistances, until 

 we reach the intersection of P; B and A C (which are here drawn 

 in). At this point of intersection, 0, the second process has done 

 all it can do for us. We see, also, that O is a fixed point within the 

 quadrilateral, since it is the intersection of the diagonals. Also, P 

 being any point, our process shows that wherever our point be 

 taken, the sum of tho distances diminishes continually as the point 



