Mr. T. S. Davies on Mr. Herapath's Demonstration. 273 



round the staff, it is out of the diagonal Hne in any other posi- 

 tion, and of course throws both the diagonal and perpendicu- 

 lar out of their true directions. 



- In measuring diagonals, &c. by means of the chain and 

 commo7i cross, Mr.Nevvton says, " The young pi-actitioner fre- 

 quently is obliged to prick his staff" in six or seven different 

 places before he can succeed in finding the diagonal, &c., but 

 in using the above he will seldom find it requisite to ground 

 his staff more than once, in order to answer the same pur- 

 pose." Now it is evident this new instrument can be placed 

 the first time only 18 inches, at most, nearer to the required 

 point than the common cross (18 inches being the length of the 

 revolving arm F H). This distance on the diagonal on either 

 side of the point where the true perpendicular falls, will cause 

 a difference in the lengths of the perpendiculars, sujiposing 

 them to be one, two, thi-ee, four, and five chains in length re- 

 spectively, of i, y^(j, y^J 2^1' ^"'^ eV P^'''^ ^^ ^^ '"*^** ^"^y' i"^ 

 their respective lengths ; and if a cross can be placed within 

 the distance of 18 inches, on the first trial, of the true point 

 where the perpendicular falls on the diagonal (as Mr. Newton 

 says it is seldom requisite to ground his staff more than once), 

 I really cannot see the necessity for pricking the staff in five 

 or six different places afterwards. Errors, I may say, of al- 

 most infinitely greater magnitude arise from inattention to the 

 correct length of the chain, and to the chain-leader not putting 

 down his pins in a perpendicular direction, &c. 



The difference of l-5th of an inch, or even of an inch or 

 two, will make little or no difference in the result, as I believe 

 it is not very usual for even professional surveyors to read off' 

 to the fraction of a link. 



XLI. Further Thoughts o« ilf;. Herapath's Demonstration. 

 By T. S. Davies, Esq. 



I'^O find an interpretation different from that upon which I 

 - had commented given to Mr. Herapath's demonstration 

 did not at all surprise me, as 1 had for some time hesitated 

 which construction to put upon it myself; and of the two 

 aspects under which I conceived it may be considered, I chose 

 that which to me appeared the most feasible, as that under which 

 tlic author himself would have it viewed. By consideruig 

 1 + 2^+ 3^ + . . .and 1 +2„ + 3„ + . . .as representations of the 

 series just deduced for integer values, ratlier than as an arbi- 

 trary denotation of the development for fractional ones (when 

 Vol. 66. No. 330. Oc/. 1825. Mm no 



