Baron of Merchiston. 167 



number 10 multiplied 56 times in succession by itself; and 

 as a thousand myriads of units make a thousand times ten 

 thousand, or 10 seven times factor, we see that the number 

 announced by Archimedes is equal to 10 multiplied by itself 

 63 times, which, even with our Arabic notation, would be 

 tedious to write, since it is unity followed by 63 cyphers. 

 But the matter becomes very simple, more simple even for us 

 than for Archimedes, if we employ the notation of the ex- 

 ponents of Descartes, which expresses only the number of 

 times the multiplication of 10 by itself ought to be per- 

 formed, for then the immense number of Archimedes would 

 be expressed in this small abridged formula, 10 63 . 



In all this, the simplicity results from the fact, that in 

 place of considering the same numbers with the multiplicity 

 of characters which express them, they are only named by 

 their rank in the indefinite progression. In following out 

 this idea, Archimedes proves that it serves equally for ob- 

 taining the products of the terms of the progression among 

 them in the simplest manner. For, suppose that we wish 

 to multiply the fourth term, which is 1000 or 10 3 , by the 

 fifth, which is 10,000 or 10 4 , the product will be 10,000,000 

 or 10 7 ; but, in place of writing all the characters which 

 compose them, it is sufficient to add together the figures 3 

 and 4, which express or expose the rank of the terms which 

 have been multiplied. For the sum 7 marks the number 

 of times that 10 is the factor in the product sought, and 

 it is sufficient to write the product 10 7 . The multiplication 

 is thus re-placed by addition, which is a much more simple 

 operation. 



Inversely, if you have the product 10,000,000 or 10 7 , 

 which is one of the terms of the series, and you wish to 

 divide it by 1000 or 10 3 , which is another term, you have 

 only to take the difference of the exponents, which is 7 less 

 3, or 4, and 10,000 or 10 4 will be the quotient sought, the 

 siiine as would be obtained by the longer process of division. 

 All the other terms of the series present the same facilities 

 for abbreviation, when they are multiplied or divided by 

 themselves, which results from the circumstance, that they 

 successively derive the one from the other in sequence, a 

 similar proportion, forming thus what is called geometrical 

 progression or by quotients, while the simpler numbers, 



