Mar on of Merchiston. 165 



Arena" who first exhibited the properties of progressive 

 numbers upon which the theory of logarithms is founded. 

 Archimedes had proposed, not the idle question of how 

 many grains of sand could be contained in a sphere equal 

 in diameter to the sphere of the stars, such as they were 

 then supposed ; but to show that a number so great, and 

 even infinitely greater might be specified and written with 

 the characters alone of numeration used in his time among 

 the Greeks. We know that these characters were letters 

 of the alphabet, which were employed consecutively in 

 their natural order, either simple or accented, to express 

 the different unions of units, tens, hundreds, until they 

 arrived at ten thousand units, which were called a myriad, 

 and which was designed by the capital letter M surmounted 

 by the alphabetic letter which expressed the number of 

 myriads to be noted. In addition to this, Archimedes con- 

 ceived an indefinite series of numbers commencing with 

 simple unity, and sucessively decupled the one by the other 

 in such a way that they were written in the order of our 

 notation, as 



1; 10; 100; 10000; 100000; &c. 

 But, as in writing them thus we are soon greatly embarassed 

 by the great number of cyphers which follow unity, we 

 abridge their expression by the assistance of an ingenious 

 method contrived by Descartes, and which consists in writing 

 only their common factor 10 affected by a numerical indicator 

 more or less considerable, which marks — which, exposes how 

 many times the common radical 10 is found multiplied by 

 itself. Then, in writing thus, the successive terms of our 

 progression, and marking below each term the rank which 

 it occupies after the first, we obtain the following lines, 

 1 ; 10 1 ; 10 2 ; 10 3 ; 10* ; 10 5 ; 10 6 ; 10 ' ; 10 3 ; &c. indefinitely. 

 1 2 3 4 5 6 7 8 

 It becomes evident at first sight that the number of the in- 

 ferior line, which expresses the rank of each term, is equal 

 to the exponent, which indicates the number of times the 

 common radical 10 is a factor in this term. This was not 

 distinct to the eye in the notation by letters employed by 

 Archimedes, and even, it was not possible for him to ex- 

 press, as we do, the character of indefinite extension, which 

 he wished to give to such a succession. 



