258 



KNO\A/^LEDGE 



[J AX. 20, 1882. 



tho SniilliBoiiian Inatitulioii, nnd tho liott litbological microHcopiHt 

 in tliiH coiintry. urnl wlni rpcontly rctunipd lo tliiH country after 

 ten yi-ars' Htiuly with I'ripf. Rmionbitiiiii nii<l others into the micro- 

 scopic chiirnctcr of rock. In finnwer to my in(|iiirie8, Prof. Hnwes 

 wrote nic this letter :— " I rend that [Miperof I'rof. Ilnhn's. lie ie a 

 kind of half-insane man, who.se iniii^ination bus run wild with him. 

 Thcae forms which he no accurutuly duiicriljus and li^'iires have lonv,' 

 been known lo exist in meteorites, and have been frequently de- 

 scribed by inineraloxists and microseoijists, Tht-y are mainly com- 

 posed of enstatito or bronzite in radial forms, and fractured in such 

 a peculiar manner as to give thoni tho appearance of structure. 

 Some of the .\merican raotcoritcs which 1 have examined show 

 these forms in ffreat beauty, but I'rof. llahn is the only man who 

 has scon anything organic in them, and his paper has excited 

 nothinjr but ridicule. It Teniiuds one of the lonj,' and laborious 

 research of a (iennan professor who found a whole flora and fauna 

 which ho named with double Latin names, and which he found in 

 his microscopic examination of basalt. It is vci-j- clear to my 

 ir.ind," continued tho Professor, " that these cranky observations, 

 viewed with tho spectacles of the imagination of Prof. Hahn, have 

 obtained more publicity than they merit." — Scientifc Amtrican. 



0m- iHatlKinntirnl €oIiimn. 



.MATIIK.M.VTIC.VL QUERIES. 



[15] — Arithmetuai. Proulem.— a and B have to build a wall 

 one hundred yards long ; A is to have five shillings a yard more 

 than B. When the wall is finished A and B each receive £50. How 

 many yards did each do, and at what rate per yard? — Hamilton- 

 Stuart. 



[This leads to an indeterminate equation. Thus, if x be number 

 of yards built by A, ;/ the number of shillings received per yard by 

 B, we have ,r(y -(• oj -Hy( 100 -j;)=. 2,000; or .T-i- 20;/ = -100.— Ed.] 



[IC] — In au old volume of problems, I came across the fol- 

 lowing: — A messenger, M, starts from \ to B (a distance a miles), 

 at a rate of i' miles per hour, but before he arrives at B, a shower 

 of rain commences at A, and at all places occupying a certain 

 distance, 2, towards, but not reaching beyond, B, and moves at the 

 rate of « miles an hour towards A. If M be caught in the shower, 

 he will have to wait until it is over. He is also to receive for his 

 errand a number of shillings inversely proportional to the time 

 occupied in it, at the rate of n shillings for one hour. Supposing 

 the distance, :, to be unknown, as also the time at which the shower 

 commenced, but all events to bo equally probable, show that the 

 value of M's expectations, in shillings, is — 



Can you or one of your readers kindly solve this problem for me 

 without using the differential calculus ? — No An.Myst. [We will 

 leave this problem for a week. "No Analyst" has set us a hard 

 one. — ^Ed.] 



[17] — ToSTOisR Problem. — A hare and tortoise have a race, the 

 hare gives the tortoise 100 yards start and runs ten times as fast ; 

 they start together, but while tho hare runs the 100 yards, the 

 tortoise has crawled ten yards ; while the hare is running the ten 

 yards tho tortoise crawls one yard; while the hare runs that one 

 yard tho tortoise has advanced -rV "f a yard ; and so on, ad infinitidii, 

 tho hare, mathematically, never overtaking tlie tortoise.— Tortoise. 

 [The parado.t was known to tho Greeks as the problem of Achilles 

 and the Tortoise. The explanation is that the distance run by either 

 animal is divided into an infinite number of parts, forming a geo- 

 metrical series, having a Unite sum. Thus the hare runs 1(X) yards 



•«• 10 -H -f. Ac. = 100 -J- (1 - Vo) = 10o( g- ) = 1 11 J yards. The tortoise 

 crawls 114 }-ards. Dividing np a finite distance into an infinite 

 number of parts does not make it infinite. — Ed.] 



[18]— AuiTHMETioAL QiKsTiON. — A man having a cask containing 

 300 gallons of wine takes out one gallon per day, putting in a gallon 

 of water to refill tho cask each day. After how many days will tho 

 mixture in the cask be half water and half wine. — YoiNc Beoin.nkk. 

 — [y. B. asks whether this ])roblem can bo solved without the aid 

 of logarithms. Nut readily. At end of one day the mixture is 

 299 _ 299 299 



„^ Uis wine, at end of second it is ^ttt. thg of -^rrr. ths wine or 



/■29n\' 



\300/ *"'"•'> *■'"! *° ""• If '" '" <lny8 the mixture in tho cask is 



half water and biilfwino, we have f — \ _-; or taking logarithin» 



of reciprocals of each side, we have *(log. 300— log. 299) = log. 2, or 



3010300 

 ItSOlr -3010300, '•=■-145^ - 207-59; or up to the 208th day 



the mixture would contain more wine than water, aft'-r tho 20t«tli 

 day it w-ould contain more water than wine.] 



.\. M. If. points out a mistake on p. 1 91, w-hcrc v/c j)iii 

 l=(v/T~t^-Hx) (v/1-t-x-r) 



Of course, this relation does not hold ; and the solution fails. We 

 had found to be the only root which could be obtained without a 

 solution for cubic or biquadratic, and as the solution, after above 

 mistake, led to root 0, wo did not (as w-c should have done), run 

 through the steps. The equation obtained just before tho mistake 

 occurs is 



l = ^/^^(v'l-^•t'•^-■r) 

 lliia will be found to give x = and the biquadnitic 

 *■*- ■W + U' + U + -l = 0. 



"T. K." considers that the proposition DM' = AM . HP, first 

 ligiiie, !>. 21 1 (ADIi a right-angled triangle, and DM jK-rp. to AB), 

 may be established without the nse of books of Euclid beyond 1. 

 and II. Thus (we rather abridge his proof) -. 



^.1/' + MB- + 2.4 J/..Ve= .4 11'= AD' + DB'^AM' + .VC + 23fX)» 

 .-.AM.MIi^MD' 

 We did not say othenvise. We said it neither is in those books, 

 nor can be given as a cofollartj on any propositions they contain. 

 Several of the jjropositions in Book III. can be as easily proved 

 from Book I., but thej' are thrown, for convenience, into a separate 

 book. 



[19] — SfM OF Squares. — I do not know if tho following simple 

 method of getting at the formula for 1' ■^ 2' -H 3' + 1= . . . . ■¥ «', 

 is new or not, but I believe it is not the usual one, and so send it on 

 the chance of your thinking it worth publishing: — 



12 3 4 5 

 112" 3 "4 5 

 1 2 L8 4 5 

 1 2 3 Ij^ 5 

 1 2 3 4 15 



Here the hoiizontal line cuts off the series-5-(n -Hi), and the zig- 

 zag line divides the figures forming the squares from the un- 

 necessary ones below. The sum of the squares cqnals .1 -H B. 



Now, i}=2C, as is readily seen from the above diagram. [Every 

 vertical column in B sums np to tw-ice the sum of a horizontal row 

 in C— En.] And „ 



B-t-C=(n-l)xY(i!+l) 



■■• B = f(»-l)x-|-(n-H) 



= -3-(n-l)(.,-H) 



.-. .•l + B = (-|--K-|(n-l)) (n + 1) 



n(n-hl)(2»-H ) 

 G 



Flore.nie E. Bo\ce. 



[9] — Several correspondents consider this problem incori-ect, in- 

 complete, or absnrd. Othei-s give incorrect solutions. It i.<, however, 

 quite correctly stated. It is correctly solved thus, or to the same 

 effect, by T. K., Charles Hammond, li. Kelley, Jas. Frobisham, K. 

 Carlson, and others. We give Mr. Hammond's solution as one of 

 the neatest in form : — 



" Since tho grass grows uniformly, its growth may be kept dovrn 

 by a certain number, independent of time, bnt varying directly as 

 the aica of the field. 



"Tho remaining horses are required to consume tho original 

 stock of grass. Their number -varies directly 08 tho area, and 

 inversely as the time. 



" Prom this it is evident that in the first case 6 horses take Iti 

 weeks to consume tho original stock of 10 acres, and G keep down 

 the grow-th. 



" Now, by proportion, the number of horses taken to consume tho 

 original stock in a lO-acro field iji G weeks is G4, and the number 

 reiiuired to keep down the growth is 2-k 



.'. Answer, 8S horses. 



N.B. For the genern.l case of this, see Newton's " Arithmctica 

 Universalis," page 90, 2nd edition, London, 1722. 



