332 



KNO\VLEDGE ♦ 



[June 1, 1883. 



they were disposBOSsed in David's pifjhlh year (2 Sam. v. 4); ro 

 that tho whole sovon books vijjoruiisly limit tlieir com[io8ition 

 within forty-povcn years, if Saul reigned forty (as stated in the 

 New Testament and best copies of Josephus). But Joshua's time 

 was then as distant as tho Wars of the Koscs are from us. 



Tho total silence of the New Testament as to tho sun miracle, 

 ospocially that of llie writer to tho lli^brows (.\i.) is most signal and 

 decisive ; and hardly less notable is that of I'hilo, wlio had many 

 occasions foralhision to it. There is a very apocryplial book (which 

 WhistoM, however, thought genuine) called the " Eighteen P.salma 

 of Solomon." It ends with this remark on the heavenly bodies : — 

 "They have not erred, from the day that Ho created them j they 

 have not left their way, from ancient generations; except when 

 God enjoined them by the command of liis servants." Now Philo, 

 in his voluminous works, often insisted on this constancy of celes- 

 tial motions; and notably in his two "Dialogues on Providence," 

 almost verbally repeats tho above, but without a hint at any 

 exception. E. L. Garbett. 



P.S. In letter 688, second paragraph, the five repetitions of 

 "p.m.," partly right and partly wrong, are the printer's fancy, as 

 I only spoke of eleven and one o'clock. 



PROBLEM IN CHANCES. 



[838] — I beg to enclose the following answer to"A. M. T.'s' 

 question [818], which appeared in Knowledge of May 18. 



If the probability of a person naming the winner in any given race 



out of ten races ? 



The probability that ho would be right inani/jjaj-h'cuZar six races, 

 and wrong in the other four (e.ij., that he would be right in the fii'St 

 six and wrong in the next four), is 



(^•(D' - i. 



But anybody would gain the prize who named the winners in any 

 eix races. 



And by elementary algebra the number of different groups of six 

 which can be obtained out of ten races is 



10.9.8.7.6.5 110 



6.5.4.3.2.1 



|6 |4 



Hence the probability of naming exactly six winners in ten races is 



|10 4^ 

 I g I , ■ j^' which is the fifth term in the expansion of the binomial 



(1 4 \'° 4^ 



— H J > and is equal to 210 — . The probability of naming at 



least six winners is 



i loA i-jg- 4^ iig. j^ i^ii 



5'o'^ 5'° + |8|2 5"'"^17|3 o'°'^|6|4 5^» 



LETTERS RECEIVED AND SHORT ANSWERS. 



OxE Open to Conviction. Considering that a series of articles 

 on "Intelligence in Animals" appeared in the first volume of 

 Knowledge, in which the view you think so unreasonable was 

 advocated on the strength of rather striking evidence, you must not 

 bo surprised to find that I take a different view. — J. de Galindez. 

 Your solution of the chance problem is not correct. You are " firm 

 as a rock " in a wrong conviction. Rocks may bo rent. As to the 

 other problem, your verbal statement of it does not agree with the 

 solution. It presents no features of interest, so far as I can see. — 

 W. G. Greener. Regret, but am quite unable to answer through 

 the post. Such letters reach me by hundreds. — Faciebat. Return 

 the letter. Cannot saj- my experience of those who write about 

 language leads me very much to admire their way of writing. 

 I doubt if they arc the best judges. When the "author of a 

 primer" tells tho author of forty books how to write English, the 

 latter is apt to feel somewhat as Charles Lamb did when a school- 

 master offered to teach him the system by which the boys of his 

 establishment were trained in composition. Your friend seeking 

 to show how a certain sentence should be worded, suggests a 

 form which would not have suited mo at all — putting "which, 

 however," for " but which." "But which" and "and which," as 

 ordinarily met with are nine times out ten incorrect ; in the tenth 

 case tho form is absolutely essential to tho sense, and tho expres- 

 sion only seems clumsy to those who have had to give so much time 

 to correcting tlie wrong use of tho expression — in school exercises, 



and so forth — tliat they cannot bear to hear it at all. I take the 

 sentence you object to, and try the effect (1) of removini; " but," 

 (2) of substituting " it " for " which," and (3) " wliich, however," 

 for " but which." In every case it fails to express my meaning. 

 As written it does express my meaning, and it is grammatically 

 sound, noting that there is an ellipsis, permissible in a sentence 

 which is meant to be a title, and is therefore reasonably presented in 

 an abridged form. Your criticism, and your friend's, reminds me of 

 one which a friend of mine made on a sentence in this form : — " If 

 a colony of Englishmen were removed to India, and there left 

 witliout interference, all would liavo died out at the end of two 

 hundred years." Jly friend insisted that either for " were " should 

 be substituted "had been," or for "would have died" shonld 

 be substituted "would die" or " would be dead" ; because he had 

 tho scholastic idea that tenses must be balanced. I objected, le- 

 cause I wanted to say what I had said, and not what the balancing 

 of tenses would have made me say — in other words, I wanted to 

 indicate, not only the state of things at the end of a certain interval, 

 but the progress of events to that end. The professed language- 

 analyser, however, will always prefer well-balanced words, tenses, 

 i-c, to so palti-y a considei'ation as the mere meaning a writer 

 wishes to convey. — An Inquirer after Tri'tii. The narrative 

 literally interpreted is, of course, absurd. What the real interpre- 

 tation may be, who can guess ? — W. Hume. Agree with you ; but 

 it is hardly worth while to have any correspondence about the 

 matter. Any man who suffers under the chimney pot for the sake of 

 appearances — may suffer. A fellow of that sort might wear a com- 

 fortable head-covering, and hold anew hat in the liand, or sling it over 

 the shoulders, or in some other way avoid the distressing suggestion 

 that he could not afford a tall hat.— Juvenis. Cannot make the "hare 

 and tortoise " matter clearer, nor answer queries such as elementary 

 textbooks deal with. As a mere detail, the earth's axis does not 

 always point to the same star. — E. C. H. Yes, we mistook ; how- 

 ever the question has now been sufficiently answered. — H. P. Any 

 elementary treatise on rigid dynamics will show you how to calcu- 

 late moments of inertia, and unless you have studied such, any 

 explanation we could give would be of no service to you.- — One who 

 Knows. As you suggest, that could hardly be discussed here. — W. 

 R. Darwin certainly does not draw any definite line between 

 varieties and species. How could he, with his views as to the 

 origin of species ? Quatrefages' other objection is absurd. One 

 might as reasonably say that the gradual addition of a blue liquid 

 to a red one would never produce a purple liquid, because the 

 various tints resulting from greater or less admixture of blue and 

 red would produce "the greatest confusion." A race which, 

 owing to changes of condition, is undergoing a change of character 

 with the progress of time, does not break up into a number of 

 races, but changes, as a whole, its progress in any direction being 

 measured by the average condition of individual members of the 

 race in regard to the changing characteristics. The constant cross- 

 ing has not the effect imagined by Quatrefages, but an effect akin to 

 the constant intermixing of the!difEerent portions of a liquid which is 

 undergoing change of tint. — Walt. Pilgrlm. Oh, favour us mth 

 an interval of repose. — H. Romeitce. You think I may have no 

 reason to complain of adverse criticism, because such criticisms, if 

 they appeared, would not be sent to me. My dear sir, ages before 

 your " Artistic and Literary Correspondence " was established, an 

 institution existed which ensured that at least every adverse criti- 

 cism should reach its mark; the institution is called "the good- 

 natured friend," to which some add, after Sheridan, a strong theo- 

 logical participle. Any one who pays you five shillings for twenty 

 adverse criticisms, every one of which would be sure to reach him 

 through the D. G. P., must have many shillings to waste, and a 

 singular taste " at that," as your country folk put it. — II. Malim. 

 Neither solution is correct, as you will presently see. 



^uv iHatJjfmatiral Column* 



GEOMETRICAL PROBLEMS. 



By Richard A. Proctor. 



Introduction. 



THE object of these hints to the solution of geometrical problems, 

 is to show the student how ho should deal with deductions 

 which are proposed to him in examination. There are many books 

 in which sets of problems are given, with several fully solved, and 

 hints supplied for the solution of others ; but these are often of 

 little use to the student. The average mathematical student re- 

 quires to learn — imt how to solve this or that problem, nor what 

 construction will help him in any particular case — Int what are the 



