16G CHARLES BABBAGE. 



aad also tables of cubes of the first ten powers of numbers reacliing to 

 100. In 1814, Professor Barlow, of Woolwich, published in an octavo 

 volume the squares, cubes, square-roots, cube-roots, and reciprocals of 

 all numbers from 1 to 1,000 — a table of the first ten powers from 1 to 

 100, and a table of the fourth and fifth powers of all numbers from 100 

 to 1,000. 



To a still greater extent were similar tables prepared on the con- 

 tinent. In France, in the year 1785, was published an octavo volume 

 of the tables of squares, cubes, square-roots, and cube-roots of all num- 

 bers from 1 to 10,0005 and in 1824 from 1,000 times to 100. A larger 

 table of squares than at that time existing was published in Hanover in 

 1810: a larger still in Leipsic in 1812 ; a more perfect one at Berlin in 

 1825; and a similar table at Ghent in 1827. 



This class of tables involves only the arithmetical dependence of ab- 

 stract numbers upon each other! To express peculiar modes of quantity — 

 such as angular, linear, superficial, and solid magnitudes — a larger num- 

 ber of computations are required. Volumes without number of these 

 tables also have been computed and published at infinite labor and ex- 

 pense. Then come tables of a special nature, of importance not inferior, 

 of labor more exacting — tables of interest, discount, and exchange; 

 tables of annuities and life insurance, and tables of rates in gen- 

 eral commerce. And then, above all others, tables of astronomy, 

 the multiplicity and complexity of which it is impossible to describe, and 

 the importance of which, in the kindred art of navigation, it would be 

 difficult to over-estimate. The safety of the tens of thousands of ships 

 upon the ocean, the accuracy of coast surveys, the exact position of 

 light-houses, the track of every shore from headland to headland, the 

 latitude and longitude of mid-sea islands, the course and motion of cur- 

 rents, direction and speed of winds, bearing and distance of mountains, 

 and, in short, everything which constitutes the chief element of interna- 

 tional commerce in modern times, dei^ends ujion the fuUness and accu- 

 racy of logarithmic tables. 



Inadequate as is the notion of the importance of these tables that has 

 been conveyed, still more inadequate must be any notice of their errors. 

 The expedients resorted to for even a limited degree of accuracy have 

 been almost innimierable. The first French Republic, aspiring to lead the 

 nations in science, undertook, through its mathematicians, by a division of 

 labor so admirable that it seemed impossible errors should be committed, 

 or, if comiaitted, remain undetected, to produce a system of logarithmic 

 and trigonometric tables so accurate that it should form a monument of 

 the kind more imposing than had ever been conceived. The attempt 

 failed, for one singular reason among others, that the computers who 

 committed the fewest errors were those who understood nothing beyond 

 the process of addition. Dr. Lardner discovered in forty tables, taken 

 at random, no less than 3,700 errata. In the Nautical Almanac Mr. Bally 

 detected more than 500 errors of calculation. The " tables requisite to 



