SQUARING THE CIRCLE 15 



immature condition of arithmetic, at the time, was the only 

 real obstacle preventing the evaluation of the ratio to any 

 degree of accuracy whatever." 



And when we remember that neither the numerals now 

 in use nor the Arabic numerals, as they are usually called, 

 nor any system equivalent to our decimal system, was 

 known to these early mathematicians, such a calculation 

 as that made by Archimedes was a wonderful feat. 



If any of my readers, who are familiar with the Hebrew 

 or Greek numbers, and the mode of representing them by 

 letters, will try to do any of those more elaborate sums 

 which, when worked out by modern methods, are mere 

 child's play in the hands of any of the bright scholars in 

 our common schools, they will fully appreciate the diffi- 

 culties under which Archimedes labored. 



Or, if ignorant of Greek and Hebrew, let them try it 

 with the Roman numerals, and multiply XCVIII by 

 MDLVII, without using Arabic or common numerals. 

 Professor McArthur, in his article on " Arithmetic " in the 

 Encyclopaedia Britannica, makes the following statement 

 on this point : 



" The methods that preceded the adoption of the Arabic 

 numerals were all comparatively unwieldy, and very simple 

 processes involved great labor. The notation of the Ro- 

 mans, in particular, could adapt itself so ill to arithmetical 

 operations, that nearly all their calculations had to be 

 made by the abacus. One of the best and most manage- 

 able of the ancient systems is the Greek, though that, too, 

 is very clumsy." 



After Archimedes, the most notable result was that 

 given by Ptolemy, in the " Great Syntaxis." He made 

 the ratio 3.141552, which was a very close approximation. 



For several centuries there was little progress towards 



