November 1, 1890.] 



KNOWLEDGE. 



257 



llcttcrs. 



[The Editor does not hold himself responsible for the opinions 

 statements of correspondents.'] 



FRACTIOXAL VALUES FOR it. 

 To the Editor of Knowledge. 



Sir, — Will you allow me to submit to your readers what 

 is, I believe, a new approximation for the number tt, 

 and which expresses the ratio of the circumference of a 

 circle to its diameter more accurately than any other 

 fractional approximation I have heard of '? 



Among the ancient Egvptians '' the ratio was taken as 

 (I)*, which is equal to 3-1(50,403, H 



The true ratio being 



3-141, .502,G53,.5H9,7!13,23H,462,043,38 . . . 

 the Egyptian value shows an error in excess of 

 •OIK, 901 ... or about one-fiftieth part of the diameter. 



Archimedes showed that tt was less than 3-^, and greater 

 than ^\^. After this achievement of the Greeks, no great 

 advance was made towards the truth until the sixteenth 

 century. The Hindus, about the year 500 a.d., used 

 filA = 3-1416 as the ratio. This is near enough for 

 all practical purposes, but is not sufficiently easy of 

 construction. 



In the sixteenth century a man named Adrian, a native 

 of Metz, gave as the value ff ;!;- = 3-141,592,9 . . . which 

 is correct to the sixth decimal inclusive. But this is not 

 much easier of construction than the Indian value. 



For all ordinary purposes the best way to draw a 

 straight line approximating in length to the circumference 

 of a circle is to take three diameters y/u.s one-fifth part of 

 the side of inscribed square. We thus obtain a straight 

 line equal to 3-141,421,356,237,3 



This shows an error in defect of -000,171,297 



or less than one five-thousandth part of the diameter. It 

 is not necessary to construct with greater accuracy, but, 

 nevertheless, whilst looking for a nearer coincidence, I one 

 day lighted upon the following : — 



Let ABODE be a circle, the diameter of which is taken 

 as unity, and AI^ the side of an inscribed square, ; BD 



is anv diameter ; l''i)= EC is drawn at viiiht ani.>les to 



BD, and meets the circumference at C. The points C and 

 D are joined by the straight line CD. 



Now if we draw a straight line equal to three times DC, 

 pim one and a third of AE, iilus six-sevenths of BD, we 

 shall approximate with marvellous accuracy to the circum- 

 ference of the circle. 



Thus : 



8 DC + "^ (AE) ^ 6 (BD) ^ 3.141,592,685,22 . . . 

 7 



showing an error in excess of -000,000,031,63 or 



less than the thirty-millionth part of the diameter. By 

 the above method, therefore, if it were possible to draw 

 with perfect accuracy, an engraver might, in a few 

 minutes, construct a straight line so nearly equal to the 

 circumference of a given circle thn rren in u (liametiT of 

 tkiitij mUcs the error iroiilil he less than the .si.rteettlh imrt of 

 an inch. _ 



It is easily seen that in the above figure 1)C= '^'^ ' 



whilst AE = 



^/2 



Thus, since \/5=2-236,067,977,5 



and x/2=l-414,213,562,4 

 6 



5 + 7 ,/^ 4. = 3-141,592,685,22 . 



I am, Sir, 



Your obedient servant, 



Gerard Daniel. 



* This appears from a i-alculation found in the Rhind papyri. 

 _A, 0. R. 



September 27th, 1890. 



To the KiUtor of Knowledge. 



De.vr Sir, — Will you inform me if there is anything to 

 be done witli a really good chronometer which has been 

 badly magnetised through a visit to some electrical works, 

 the proximity to the powerful magnets there employed 

 causing an error of several seconds daily which cannot be 

 rectified ■? — Yours truly, 



C. Parkinson. 



[The watch may be demagnetised by placing it in a 

 magnetic field produced by a coil through which a power- 

 ful alternating current is passing. The intensity of the 

 correcting magnetic field should be at least as great 

 as that which originally caused the derangement, and 

 the watch should be slowly withdrawn from the coil 

 through which the alternating current is passing. Mr. 

 Common, in the Ohacrrdtori/ for November 1889, gives an 

 account of the demagnetisation of a chronograph watch 

 which he had badly magnetised by going too near to a 

 dynamo at work, so that from a rate of fifteen seconds 

 ga,ining, a losing-rate of from five to six miiuitcs a day had 

 been caused. He took it to the chief engineer of the 

 electric lighting works of Taddiugton Station, who placed 

 it within a coil used tor the purpose, through which an 

 alternating current of about 140 volts was passing, and 

 turned it about slowly, withdrawing it while doing so. 

 The result was perfectly successful, and the watch at 

 once returned to its old rate of fifteen seconds gaining.— 

 A. C. R.J ____ 



THE WORD •• BROAD." 

 To the Ktlitor 0/ Knowledge. 

 Dear Sir,— The great English Dictionary now in course 

 of publication is not only admirable for the abundance of 

 information it contains, but also for the care with which 

 its able Editor keeps within the limits of ascertained fact. 

 One instance of this is found under the word lUontl, the 



