each quantity in the left-hand column being the square root of the one above it, and each quantity in the right-hand column being the half of the one above it. To construct this table Briggs, using about thirty places of decimals, extracted the square root of 10 fifty-four times, and thus found that the logarithm of 1.00000 00000 00000 12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of this form (i.e. for numbers beginning with 1 followed by fifteen ciphers, and then by seventeen or a less number of significant figures) the logarithms were proportional to these significant figures. He then by means of a simple proportion deduced that log (1.00000 00000 00000 1) = 0.00000 00000 00000 04342 94481 90325 1804, so that, a quantity 1.00000 00000 00000 x (where x consists of not more than seventeen figures) having been obtained by repeated extraction of the square root of a given number, the logarithm of 1.00000 00000 00000 x could then be found by multiplying x by .00000 00000 00000 04342.... To find the logarithm of 2, Briggs raised it to the tenth power, viz. 1024, and extracted the square root of 1.024 forty-seven times, the result being 1.00000 00000 00000 16851 60570 53949 77. Multiplying the significant figures by 4342 ... he obtained the logarithm of this quantity, viz. 0.00000 00000 00000 07318 55936 90623 9336, which multiplied by 2^47 gave 0.01029 99566 39811 95265 277444, the logarithm of 1.024, true to 17 or 18 places. Adding the characteristic 3, and dividing by 10, he found (since 2 is the tenth root of 1024) log 2 = .30102 99956 63981 195. Briggs calculated in a similar manner log 6, and thence deduced log 3. It will be observed that in the first process the value of the modulus is in fact calculated from the formula. h 1 -------- = ---------, 10^h - 1 log(e) 10 the value of h being 1/2^54, and in the second process log10 2 is in effect calculated from the formula. 1 2^47 log(10) 2 = [2^(10/2^47) - 1] × --------- × ----. log(e) 10 10 Briggs also gave methods of forming the mean proportionals or square roots by differences; and the general method of constructing logarithmic tables by means of differences is due to him. The following calculation of log 5 is given as an example of the application of a method of mean proportionals. The process consists in taking the geometric mean of numbers above and below 5, the object being to at length arrive at 5.000000. To every geometric mean in the column of numbers there corresponds the arithmetical mean in the column of logarithms. The numbers are denoted by A, B, C, &c., in order to indicate their mode of formation. Numbers. Logarithms.