278 M. Biot's Abstract of Mr Napier'^s 



medes is equal to 10 multiplied by itself 63 times — a number 

 which, even with our Arabic notation, is a long one to write out, 

 being unity followed by 63 zeros. But this matter becomes 

 very simple, and much more so for us than for Archimedes, if 

 we employ the Cartesian method of exponents, which simply 

 expresses how many times the multiplication of 1 by itself has 

 entered into the operation, for then that immense number of 

 Archimedes is written under this little contracted form, 10^^. 



In all this the simplification results from the fact that, instead 

 of considering the numbers themselves with the multiplicity of 

 characters which express them, we merely designate them by 

 their rank in the indefinite progression, and to express that rank 

 is always by much the shorter process. In following out this 

 idea, Archimedes proves that it is equally available for obtaining, 

 by a very easy process, the products of terms in the progression 

 multiplied together. For example, suppose we wish to multiply 

 the fourth term, which is 1,000, or 10^, by the fifth, which is 

 10,000, or 10\ the product will be 10,000,000, or lO^; but 

 instead of thus seeking it, and painfully writing out the com- 

 ponent characters, it will suffice to add together the figures 3 

 and 4, which express (are exprnients of) the rank in the progres- 

 sion, of the two terms which are to be multiplied together. For 

 their sum, 7, marks the number of times 10 is factor in the 

 product sought for, and thus enables us at once to write out the 

 product 10^. Thus, multiplication is superseded by addition a 

 much more simple operation. Inversely, if this term in the 

 series, 10,000,000, or 10"^, be given, and we are required to 

 divide it by the other term 1,000, or 10^, we need only take the 

 difference of the exponents, which is 7 minus 3, that is to say, 4 ; 

 and 10"^, or 10,000, will be the quotient sought for, the same 

 which would have been obtained tediously by division. All the 

 other terms of the indefinite series offer the same facility of 

 abbreviation when required to be multiplied, or divided, together; 

 which results from this, that they are derived successively the 

 one from the other by an unvarying ratio, thus forming what 

 we call a geometrical progression^ ov by quotients; while, on 

 the other hand, the more simple numbers which express the rank 

 of each term, increasing simply by one unit, and always one 

 unit, in passing from one term to the next, constitutes another 



