TRACT 21.
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idea of this matter, and that he altered the word artificials tologarithms in his first book, on the description of them, whenhe printed it, in the year 1614, and that he would also havealtered the word every where in this posthumous work, if hebad lived to print it: for in the two or three pages of appen-dix, annexed to the work by his son, from Napier’s papers,he again always calls them logarithms. This appendix relatesto the change of the logarithms to that scale in which 1 is thelogarithm of the ratio of 10 to 1, the logarithm of 1, with orwithout ciphers, being 0; and it appears to have been writtenafter Briggs communicated to him his idea of that change.
Napier here in this appendix also briefly describes somemethods, by which this new species of logarithms may beconstructed. Having supposed 0 to be the logarithm of I,and 1, with any number of ciphers, as 10000000000, thelogarithm of 10; he directs to divide this logarithm of 10,and the successive cjuotients, ten times by 5 ; by which divi-sions there will be obtained these other ten logarithms, viz-2000000000, 400000000, 80000000, 16000000, 3200000,640000, 128000, 25600, 5120, 1024: then this last logarithm,and its quotients, being divided ten times by 2, will give theseother ten logarithms, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1.And the numbers answering to these twenty logarithms, weare directed to find in this manner; namely, extract the 3throot of 10, with ciphers, then the 5th root of that root, amiso on, for ten continual extractions of the 5th root; so shallthese ten roots be the natural numbers belonging to the firstten logarithms, above found in continually dividing by 5 :next, out of the last 5th root we are to extract the squareroot, then the square root of this last root, and so on, for tensuccessive extractions of the square root; so shall these lastten roots he the natural numbers corresponding to the loga-rithms or quotients arising from the last ten divisions by thenumber 2. And from these twenty logarithms, 1,2, 4, 8, 16,&c, and their natural numbers, the author observes that otherlogarithms and their numbers may be formed, namely, byadding the logarithms, and multiplying their corresponding