238 



K N:0 W L E D G E . 



[October 1, 1890. 



in it a regular heptagon. He shows that although the 

 mouth of the last tunnel does not emerge exactly at the 

 mouth of the first tunnel, it nevertheless emerges within 

 fiftj' feet of it. 



While considering this matter T accidentally came 

 across a solution of the problem, which, although it lacks 

 mathematical accm-acy, is marvellously near to the truth, 

 while, at the same time, it is incomparably more simple 

 in construction than Rober's method. 



If the diameter of the circumscribed cu'cle is unity and 

 .1- is the side of an equilateral hepta,gon, we have 



64.i-«-112..-*+56..'-=7 

 from wliich we obtain for the value of .r 



•433,883,739,117,558,120,475 . . . 

 Now the fraction 63—91/7 



61- 



•433,883,738,080, 



-70^3-65^2 

 and the accuracy is so great that if a regular inscribed 

 heptagon were constructed in this manner, even in a circle 

 a thousand miles in diameter, the error of each side (and 

 the error, as will be observed, is in defect) would be less 

 than the jifteenth part of an inch. 



The simplest form of the above fi-action, for construc- 

 tion, would be : 7— ^/l 



a fraction the parts of which are easily constructed. 



Probably the easiest way to construct the square root of 

 seven is the following : 



ABC is any circle, the 

 diameter of which is taken 

 as the unit of reetUiueal 

 measurement. 



Draw any diameter OH, 

 and in it take 



8 



Through F draw the 

 straight line DFE at right 

 angles to OH, and termi- 

 nated by the circumference 

 of the cu^cle at the points D and F. 



Then DE=^ 



4 

 I am. Sir, Your obedient servant, 



August 23rd, 1890. Geeakd D.ajslel. 



[There is a very old and comparatively simple way of 

 approximately di\iding a circle into seven parts, which De 

 Morgan, in his Xotes on the History of Persiwtiie, pub- 

 Hshed in the Athenaum (Sept. 12, 1863) says that he had 

 traced through writers on perspective up to Albert Diirer, 

 but that he could not trace it any further back. Albert 

 Diirer assumed the side of a regular heptagon inscribed in 

 a circle to be approximately equal to half the hne which 

 joins the two intersections of the circles in Euclid's first 

 proposition. That is, if the diameter of the cu-cle is 4, 

 the side of the heptagon is approximately V^d. De 

 Morgan quaintly says of this : "It is too small, but any- 

 one who would feel satisfied with £1 as composition for a 

 debt of f 1 Os. O-Jd. ought to be a trifle better satisfied with 

 Albert Diirers heptagon.'" An error of less than one inch 

 in forty feet is good enough for any ordinary drawing pur- 

 poses. Mr. Daniel's approximation about corresponds to 

 an error of a farthing in a million pounds, or to an error 

 of an inch and a half in the meeting of Sir Wm. R. 

 Hamilton's tunnels at the earth's equator. — A. C. R.] 



To the Fditor of Kxowledcje. 



Deak Sir,— The titles of Figs. 10 and 11 illustrating 

 my paper on " Binary Stars of Short Period," in the 

 August number of Knowledge, were by some oversight 

 interchanged. Fig. 10 should be apparent orbit of 85 

 Pegasi, and Fig. 11 apparent orbit of ^ Scorpii. The dia- 

 grams themselves are correct. 



Yom's truly, 



J. E. Gore. 



NUMBERING THE DUST OF THE AIR. 



By Dr. McPhersox, F.R.S.E. 



ONE of the most remarkable contrivances of modern 

 times enables us to count the minute inorganic 

 dust-particles in the air. To Mi-. John Aitken, 

 an ingenious Scotch physicist, we owe this new 

 method of research. 

 The bright motes that dance in the sunbeam seem 

 bejond the power of computation, yet, by a marvel of 

 mechanical ingenuity, Mr. Aitken has counted them. I 

 shall never forget my rapt astonishment the day I first 

 counted the dust in the Lecture Room of the Royal 

 Society of Edinbm^gh, with his instrument and under his 

 dii-ection. The in\-isible particles in the air were brought 

 within the range of vision, and even within the limit of 

 easy enumeration. 



The method of numbering the inorganic particles in the 

 ail- depends upon a pi'inciple which was established by 

 ilr. Aitken in his determination of the foiTnation of fogs. 

 He showed that wthout dust there could be no fogs, no 

 mist, no raiu. Without dust there would be only dew on 

 the grass and road. This principle can be easily illus- 

 trated. Let common air be forced through a filter of 

 cotton-wool into a glass receiver, from which the ak has 

 been exhausted ; and let a glass receiver, fiUed with 

 common au-, be placed beside it. If steam be now ad- 

 mitted into the receivers, the one containing the common 

 dusty air will soon be dense with fog, while the other con- 

 taining the pure filtered ak wUl remain perfectly clear. 

 The particles of dust, then, are the fi'ee- surfaces which, in 

 certain conditions, attract the water-vapom- of the atmo- 

 sphere to form fog. Invisible before, they are touched by 

 the magic wand of a loweruig temperatm-e, and start into 

 visible existence ; the dust-particles are clothed all over 

 with the moistm-e, and become fog-particles. 



It then occurred to Mr. Aitken that if a small measured 

 quantity of the common dust- impregnated air be mixed in 

 a receiver with a large measured quantity of dustless air 

 (which has been filtered through cotton-wool), the particles 

 of dust would be some distance from each other ; and, 

 when these particles were made centres of condensation of 

 vapour by lowering the pressure, fog-particles would be 

 formed, which could be counted by means of a magnifying- 

 glass. If, moreover, these particles fell from a certain 

 height on a small, measured area, the number could be 

 accurately ascertained. That is the secret ! 



Though Mr. Aitken has made great improvements on 

 his instrument, the principle is the same, and the first 

 apparatus is most easily explained without the assistance 

 of diagrams. Into a common glass flask, of carafe-shape, 

 and flat-bottomed, of 500 cubic centimetres (about 32 cubic 

 inches) capacity, are poured 50 c.c. of distilled water. 

 Through the air-tight stopper are passed two small tubes, 

 at the end of one of which is attached (a little to the side 

 of the orifice) a small square silver table of one square 



