172 report— 1878. 



and pulling up pile stumps in deep water, in which latter operation it is 

 very successful, and three or four pile stumps can be drawn at one effort 

 by attaching chains hanging from the ceiling of the bell chamber to the 

 heads of the piles, and then raising the pile by its hoisting chains, which 

 have a surplus working strength of about seventy tons when the bell is 

 under water. 



Report of the Committee, consisting of Professor Cayley, F.R.S., 

 Professor G. G. Stokes, F.R.S., Professor H. J. S. Smith, F.R.S., 

 Professor Sir William Thomson, F.R.S., Mr. James Glaisher, 

 F.R.S., and Mr. J. W. L. Glaisher, F.R.S. {Secretary), on Ma- 

 thematical Tables. 



[Plate IV.] 



Account of the Calculation of the Factor Table for the Fourth Million. 



A description of the different factor tables that have been .published 

 is given in the British Association Report, 1873, pp. 34-40 ; and a more 

 complete historical account of factor tables, especially of Felkel's and the 

 manuscript tables of the last century is contained in the ' Proceedings of 

 the Cambridge Philosophical Society,' vol. hi. part iv. pp. 99-138, 1878. 

 It is only necessary, therefore, to give a brief notice of the extensive 

 tables that have been published during the present century, and which 

 it is the object of the Committee to complete. 



These tables are : — 



(1). Chernac's Cribrum Arithmeticum, which gives all factors of all 

 numbers not divisible by 2, 3, or 5 from 1 to 1,000,000. 



(2). Burckhardt's Table des Diviseurs, which gives the least factor of 

 all numbers not divisible by 2, 3, or 5 from 1 to 3,036,000." 



(3). Dase's Factoren Tafeln, which give the least factor of all num- 

 bers not divisible by 2, 3, or 5 from 6,000,000 to 9,000,000. 



The reason of the gap between 3,036,000 and 6,000,000 is as follows : 

 — Burckhardt completed the publication of his three millions in 1817, and 

 some time previous so 1849 Crelle presented to the Berlin Academy the 

 manuscript of the factor tables for the fourth, fifth, and sixth millions. 

 In 1850 Gauss urged Dase to calculate factor tables for the seventh, 

 eighth, ninth, and tenth millions, as the three intermediate millions 

 were in the possession of the Berlin Academy, and he did not doubt 

 that sooner or later they would be published. In 1860, through the 

 support of friends in his native town, Hamburg, Dase, who was 

 distinguished for his ability in calculation, was enabled to devote him- 

 self wholly to the carrying out of Gauss's project. On September 

 11th, 1861, he died suddenly, leaving the seventh million complete and 

 the eighth million nearly complete; he had also determined a great 

 number of the factors for the ninth and tenth millions. Dr. Rosen- 

 berg, of Hamburg, undertook the continuation of the work, and the 

 seventh million was published at Hamburg in 1862, the eighth in 

 1863, and the ninth in 1865. In the preface to the ninth million it is 

 stated that the tenth million was near completion. There was thus left 



