40 



• KNOWLEDGE . 



[July 11, 1884. 



same day at VancouTer Island, and 11m. 38m. p.m. at Wellington, 

 New Zealand. Sliips returning from circumnavigation of the 

 world of course change their day on coming to the first ]Jort witiiin 

 these meridians. — William Wigan. I regret, that I am powerU'ss to 

 help yon. — Drum sends a newspaper cutting of the score of a cricket 

 match between two towns named Lenno.ivale and Adelaide. In 

 the first innings Lennoxvale scored 2 and Adelaide 4. The 

 second was not played oat. Seemingly the wicket soon 

 ceased from troubling, and the weary were at rest. — 

 Phobo.«. Depend upon it our contributor knows most thoroughly 

 what he is writing about. I cannot conscientiously recommend 

 you a cheap instrument. Neither of the gentlemen you name ever 

 made a telescope in his life. I am quite uncertain about coming 

 to town. — Julia Usher. No telescope of less than 2J inches in 

 aperture is of the slightest use for astronomical i)urpo6es ; and the 

 lowest price at which a first-class instrument of this size is pro- 

 curable is seven guineas, or thereabouts. 1 cannot undertake to 

 recommend individual makers, of whom, by the way, there are very 

 few; though sellers of telescopes abound. — O. P. Asa really ad- 

 mirable introduction to the subject, nothing surpasses the 

 " Marvels of Pond Life," by Mr. H. J. Slack, published by Groom- 

 bridge <fe Son, London. For an exhaustive account of Cyclops, 

 Daphnia, &c., see Dr. Baird's " Natural History of the British Ento- 

 mostraca," published by the Ray Society. The " Micrographic 

 Dictionary," published by Van Voorst, contains a mass of informa- 

 tion concerning microscopic objects of every description — animal, 

 vegetable, and mineral. For the adequate study of Infusoria 

 proper, the " Manual of the Infusoria," by Mr. W. Savill, Kent, 

 published by Bogue, London, must be consulted. The last two 

 works named are both bulky and somewhat costly.— Gf.bty Girtox. 

 I do not insert your jen d' esprit, just to show that I can resist 

 temptation; but your P.S. put an amount of pressure upon 

 me almost too great for even an editor to withstand. — Aha 

 Heather Bigg. Your able paper unfortunately occupies more space 

 than I could devote to a subject of comparatively limited popular 

 interest.^jAMES Gillespie. My mere silence might have sufficed 

 to indicate what I " think of " your pamphlet. I stand in a species 

 of dumb amazement at any one who is so hopelessly ignorant of 

 the most elementary facts of the subject, presuming to dogmatise 

 upon them. I should no more dream of arguing with you than I 

 should with a gentleman in a straight waistcoat who alleged that he 

 was the Podasokus, or little Oozly bird. Any child of eleven who 

 knows the principal constellations will tell you that we do not see 

 the same stars in the September night sky that we do in the March 

 one. Your prose is bad, your " poetry " verse and verse. As for 

 letting you " know what you should do to bring it out," I would 

 implore you, if you value your reputation, to keep it in, with all your 

 might and main. — W, Hill & Son send a description of a " Hygienic 

 Bakehouse," which would appear to present a delightful contrast 

 to the filthy underground cellars in which Londoners have been 

 from time to time shocked to learn that some of their bread is 

 made.^W, P. Thompson Boixt. Your " New Patent Convention " 

 received, but its great length precludes its insertion. — Rev. T. W. 

 Webb. Received. — John A. Stewart sends an account of the 

 rescue of a man from a mine after three weeks' imprisonment, in 

 consequence of a woman dreaming that he was immured in it. His 

 entombment was quite unsuspected otherwise. This happened forty 

 years ago. — T. A Vligk (sic) takes exception to the statements on 

 p. 9 as to the danger to be apprehended from disease germs. — J. F, O. 

 I can only regret that the very great length to which your com- 

 munication on " Fortune-Telling extends prevents its insertion, and 

 I scarcely see ray way to curtail it without destroying the sequence 

 of your argument. 



(Bm iHatDtmatiral Column. 



NOTES ON EUCLID'S SECOND BOOK. 

 By Richard A. Pkocior. 



IN the second book Euclid deals with the relations between the 

 rectangles contained by straight lines and the parts into which 

 they may be divided. The method he adopts is somewhat cum- 

 brous — so far, at least, as Problems 2-10 are concerned. The 

 student must not deal with problems on the second book in Euclid's 

 manner. In order to illustrate the proper method of dealing with 

 such deductions we give new solutions of Propositions i to 10, pre- 

 mising that Propositions 2 and 3 are particular cases of Prop. 1, 



Euc. II., Prop. 4 should be thus established : — 



Let AB be the given A + b 



straight line divided into *- 



any two parts in the point C : the square on A B shall be equal to 



the squares on AC, C B, together with twice the rectangle A C, C B. 



By Prop. 2 the sijuare on A B is equal to the rectangle A B, A C, 

 together with the rectangle A B, B C. 



But by I'rop 3, the rectangle A B, A C is equal to the rectangle 

 A C, C B, together with the square on A C ; and the rectangle A B, 

 B C is equal to the rectangle A C, C B, together with the square 

 on BC. 



Hence the square on A B is equal to twice the rectangle A C, 

 C B, together with the stiuares on A C and C B. 



Cor. — If a straight line be divided into two equal parts the 

 s<|uare on the whole line is equal to four times the square on either 

 half. 



Prop. 4 may be enunciated thus, If a straight line he divided into 

 any two partSj the square on one part in less than the square on the 

 whole line by twice the rectangle cmtuined by the parts together 

 with the square on the other part. 



Prop. 5 should be established ^__ 



thus; let the straight line AB A J 5 i3 



be divided into two equal parts 



in C, and into two unequal parts in D ; then the rectangle A D, 



D B together with the square on C D shall be equal to the square 



onCB. 



Since A D is made up of A C, C D whereof A is equal to C B, 

 the rectangle A D, D B is equal to the rectangle C D, D B together 

 with the rectangle C B, D B (Euc. 11., 1): that is, to twice the 

 rectangle C D, D B, together with thi' square on D B (Prop. 3). 

 Add the square on C 1). Then the rectangle A D, D B together 

 with the square on C D, is equal to twice the rectangle C D, D B 

 together with the squares on C D, U B ; that is, to the square on 

 C B (Prop. 4). 



Cor. Since the square on \ C or C B is equal to the rectangle 

 A C, C B, it follows that if a straight line is divided into unequal 

 parts the rectangle contained by these is less than the rectangle 

 contained by the halves of the line ; and also (the deficiency being 

 the square on C D) that the more unecjual the parts the smaller is 

 the rectangle contained by them. 



Prop, ti should be established thus, — 



Let the straight line A B be bisected in C, and produced to D ; 



then the rectangle 



E A S B u A D, D B together 



with the square on 

 C B, shall be equal to the square on C D. 



Produce D A towards A to E, making E A equal to B D, so that 

 C E is equal to C D and B E to A D. Then by Prop. 5, the rect- 

 angle E B, D B together with the square on C B is equal to the 

 S([uare on C D ; that is, the rectangle A D, D B together with the 

 square on C B is equal to the square on C D. 



Props. 5 and 6 may be included in one enunciation thug — The 

 rectangle contained by the sum and difference of two straight lines is 

 equal to the difference of their squares (A C, C D being the two lines 

 referred to) ; or thus^ taking AD andh V as the two lines of reference, 

 the rectangle contained by two lines is equal to the square of half 

 their su7n diminished hy the square of half their difference. 



Prop. 7 is proved thus : — 



Let the straight line AB 



be divided into any two* — ^ B 



parts in C ; the squares on 



AB, BC, shall be equal to twice the rectangle AB, BC, together 



with the square on A C 



The square on A B is equal to the squares on A C, C B, with 

 twice the rectangle A C, CB (Prop. 4). Add the square on C B. 

 Then the squares on A B, C B are together equal to the square on 

 A C, twice the square on C B and twice the rectangle A C, C B ; 

 that is to the square on A C together with twice the rectangle A B, 

 BC(Prop. 3). 



Prop. 7 may be enunciated thus : — The square on the difference of 

 tico lines (ABandBC) is less than the sum of the squares on those 

 linesby tn-ire the rectangle contained by thetii. 



Prop. 8 is proved thus : — 



Let the straight line A B a c b d 



be divided into any two ' ' ' ' 



parts in C; then four times the rectangle A B, B C together with 

 the square on A C is equal to the square on the straight line, made 

 up of A B and B C together. 



Produce A B to D making B D equal to B C, so that A D is the 

 line made up of A B and B C together. Then the square on A D is 

 equal to the squares on A C, C D together with twice the rectangle 

 A C, C D. But the square on C D is equal to four times the square 

 on C B ^Prop. 4, Cor.) and the rectangle A C, C D is equal to twice 

 the rectangle A C, C B (Prop. 1). Hence the square on A D is equal 

 to the square on A C, together with four times the square on C B 

 and four times the rectangle A C, C B ; that is, to fotir times the 

 square on A C and four times the rectangle A B, C B (Prop. 3), 



Cor. — If AC is equal to B C and therefore to C D, we have the 



