July, 1913. 



KNOWLEDGE. 



249 



the microscope ready, and to give some idea of how 

 many of them there are we show in Figure 262 the 

 more important portions of an H Edinburgh 

 Student's Microscope and the separate pieces of 

 which they are made up. It may surprise some of 

 our readers to learn that there are sixty-four of 

 these and ninety-six screws in the stand alone. 



To assemble a microscope is the term used for 

 putting the stand together — that is, combining the 

 stage, foot, limb, and mirror (see Figure 263). After 

 this the instrument has to go through the testing 

 room examination (see Figure 264), when it is ready 

 to leave the works and be fitted with such eyepieces 

 and objectives as its future owner needs for his work. 



CORRESPONDENCE. 



QUADRATURE OF THE CIRCLE. 

 To the Editors of " Knowledge." 



Sirs, — In " Knowledge " for November last you published 

 a letter from me touching the quadrature of the circle. In 

 this letter I asked certain questions on the subject. I also 

 stated that I had ascertained by geometry that the perimeter 

 of a circle is equal to a certain triangle described on same. 



Up to date no reply to the questions referred to has 

 appeared in " Knowledge," and I am still holding in 

 abeyance the geometric solution referred to pending satisfac- 

 tory answers from some source to these queries. 



What I wish to be informed on is : — 



1st, Has the solution referred to above ever been published or 

 made known before it appeared in " Knowledge," i.e.. 

 either in ancient or modern works ? 



2nd, Has it ever been satisfactorily proved or disproved in 

 any way by any author ? 



One gentleman did write to you on the subject, but he 

 made no reference whatever to these questions, although he 

 criticised the solution. 



In reply to this writer I give Figure 265, showing that the 

 solution is exact and not approximate only. 



Arithmetic of the Quadrature. 



I also send Figure 266 for the purpose of showing that my 

 geometric solution referred to above can be verified by 

 arithmetic. 



Attention may be drawn to the following points : — 

 1st, The whole of the ratios given are based on the square 

 root of 5 and the square root of 1 (which, of course, is l),and 

 that they are therefore commeasurable. 



2nd, The ratio of the circumference of the circle to its 

 diameter is identical with the ratio of — 



(a) The area of the outer circle, minus the square to 

 (6) The area of the square, minus the inner circle, i.e., 

 the ratio of (a) to (6) or of E to D in the diagram. 



Table 46 shows the results of using any different ratio from 

 v^X v'l: 



Figure 265. 



By ordinary geometry the square C is = A and B together. 

 Let diameter = 1. 

 The square A = 1. 

 And square B = -25. 

 And therefore square C = 1 - 25. 

 Also square on E D F will = 5 (1-25 + 4). 

 And square on E F will = 1. 



Therefore E I) F = v 5 and EF= \ 1. 



And therefore E D F E = \ 5 X \ 1 . 



A The inner circle-area v'5 + v'l + -25 



(i radius). 

 B The square-area v 5 X \ 1 + E -5- 2. 

 C The outer circle-area v'5 X v'l + 



v 2 + (-25 + \ 2). 

 D The four spaces D singly or together. 

 E The four spaces E singly or together. 



Figure 266. 



Ratios (taken inversely, etc.) : 

 Circumference to diameter ... v 5 X \ 1 



E to D v 5 X \ 1 



The square to the lesser circle v 5 X \ \ — 2 

 The greater circle to the square v 5 X \ 1-^-2 



A to E •5X V 'l+2-' 



A to D v 5 X v'l X v'l 



Table 46. 



These figures (and any further calculations that may be 

 made on the same lines) show conclusively that the figures of 

 the ratio " ^5X1 "are the most " symmctral " that can be 

 produced in conjunction with Figure 266. 



Therefore, if they are not the true figures of the circle, 

 there must be some other figures or polygon (regular or 

 irregular) to which they do apply ; and it will also follow that 



this other figure is a more symmetrical figure than the circle — 

 which is impossible. 



On the strength of what I have advanced above, I hope 

 some scholar will now answer the questions given above. 



Brisbane, 

 Queensland. 



GEOMA. 



