On a Property of the Interminable Decimals. 15 



It is perhaps needless to remark that air-holes are left 

 through the upper part of the sand of the mould to allow a 

 free escape of the air upwards, as it is displaced by the metal 

 entering by the ingate. 



My apology for writing in so much detail on such a subject 

 is, that I am well-aware the number of amateur telescope- 

 makers in this country is much greater than generally ima- 

 gined ; and one who immediately published his discoveries 

 tor the benefit of brother amateurs, ought also to caution 

 them against useless and unphilosophical methods. 



IV. On a property of the Interminable Decimals iiohich arise 



from the development of an Incommensurable Fraction, ^ By 



J. R. Young, late Professor of Mathematics^ Belfast^ jjj,i, 



THE following curious property of fractional developments 

 has, I believe, hitherto escaped notice. I find that, in 

 at least one interesting inquiry, it is capable of a practical ap- 

 plication, not altogether unworthy of notice : whether or not 

 it can be turned to useful account in any other direction, I 

 am not able to say ; but I think it deserving of record, as a 

 remarkable arithmetical theorem, though depending only upoii 

 a very simple principle. Before explaining this principle, it 

 may be better first to exemplify the property itself, as a thing 

 ceases to appear remarkable, and often ceases to be valued, 

 as soon as it is found to be capable of a simple explanation. 

 Let an incommensurable fraction be taken at random, saj^ 



-; then ?' 



^= •142857142857142S571428, &c. 



'"It is a property of the development of such a fraction that^ 

 stop wherever we may, a certain number of figures may be 

 cut off the end, such that if the figures left be all multiplied 

 by a certain determinable number, all the original figures will 

 be reproduced, with the exception of a limited number of the 

 leading figures. In performing the multiplication here men- 

 tioned, account must, as usual, be taken of the carryings from 

 the figures cut off. 



Cut off one place from the foregoing decimal, and multiply 

 the whole row, omitting that one, by 3 ; arranging the result 

 underneath;^ so that the like reproduced figures may stand 



• Communicated by the Author. - .>":"^*Art^ 



