Utilizing Compact Logarithmic Tables. 



225 





3° 0'. 



7° 33 r . 



15° 6'. 



Sine 

 8-71. 



Msinl"eot.r. 



Sine 

 9118. 



Msiul"cot#. 



Sine 

 9-415. 



Msinl"cot.r. 







1 



2 

 3 

 4 

 5 



6 



7 



8 



9 



10 



11 



12 



880 01637 

 884 03375 

 888 05076 

 892 06740 

 896 08367 

 900 09956 

 904 11510 

 908 13024 

 912 14503 

 916 15944 

 920 17347 

 924 18714 

 928 20042 



5-6039633 



Error. 

 +2 

 1 

 2 

 3 

 3 

 4 

 3 



56 67460 



58 26316 



59 85166 

 61 44010 



63 02818 



64 61681 



66 20507 



67 79328 



69 38143 



70 96952 

 72 55756 



74 14554 



75 73346 



5-2010109 



Error. 



+ 1 

 1 

 2 

 3 

 3 

 5 

 6 

 7 



81 52442 



82 30475 



83 08507 



83 86537 



84 64567 



85 42594 



86 20621 



86 98646 



87 76669 



88 54691 



89 32712 



90 10731 

 90 88749 



4-8922838 



Error. 



+ 1 

 1 

 2 

 3 

 3 

 4 

 5 

 6 

 7 



It will be obvious from the examples given, that we can for any 

 number, whose significant figures do not exceed seven more than 

 the tabular number, find the quantity required to complete its 

 logarithm. It hence follows that if we can by any particular 

 process find a larger number of preceding figures, or larger value 

 for x, we shall always have it in our power to extend the value 

 so found to seven additional places, thereby advancing beyond 

 the ten places to which we have hitherto limited ourselves. This 

 may be done by determining factors whose product shall be 

 nearly identical with the first eight digits of the given number. 



Let this given number be 15358979323846. Point off the 

 first eight significant figures, and subtract them from the next 

 greater complete square number, and find the remainder; to 

 this remainder add double the root of the square, increasing this 

 double by a single unit ; afterwards add continuously the series 

 of odd numbers which next follow that double root so increased, 

 until a succession of sums be obtained to such extent as may be 

 desirable. 



3920 2 = 15366400 Next greater square. 

 15358979 



7396 = 86 2 ... 7421 



7841 ... 2r+l, add 



15376 =124 2 .. 15262 



7843 Next odd No. 



23104 =152 2 ..23105 



7845 ... &c, 



30976 = 176 2 .. 30950 



7847 ... &c, 



388C9 = 197 2 ..38797 

 Phil Mag. S. 4. Vol. 30. No. 202. Sept. 18G5. Q 



