as to which numbers were "perfect" or why. A view that was shared by Plato held that 10 was a perfect number. Mathematicians, including the mathematician-philosopher Pythagoreans, proposed as a perfect number, the number 6. The number 10 was thought perfect because there are 10 fingers to the two hands. The number 6 was believed perfect for being divisible in a special way: a sixth part of that number constitutes unity; a third is two; a half — three; two-thirds () is four; five-sixths (pentamoiron) is five; six is the perfect whole. The ancients also considered 6 a perfect number because the human foot constituted one-sixth the height of a man, hence the number 6 determined the height of the human body. Thus both numbers, 6 and 10, were credited with perfection, both on purely mathematical grounds and on grounds of their relevance in nature. Belief in the "perfection" of certain numbers survived antiquity, but this quality came to be ascribed to other numbers as well. The perfection of the number 3 actually became proverbial: "omne trinum perfectum" (). Another number, 7, found a devotee in the 6th-century Pope Gregory I (Gregory the Great), who favored it on grounds similar to those of the Greek mathematicians who had seen 6 as a perfect number, and in addition for some reason he associated the number 7 with the concept of "eternity." The Middle Ages, however, championed the perfection of 6: Augustine and Alcuin wrote that God had created the world in 6 days because that was the perfect number. The Greek mathematicians had regarded as perfect that number which equals the sum of its divisors that are smaller than itself. Such a number is neither 3 nor 7 nor 10, but 6, for 1 + 2 + 3 = 6. But there are more numbers that show this property, such as 28, which = 1 + 2 + 4 + 7 + 14. It became customary to call such numbers "perfect." Euclid gave a formula for (even) "perfect" numbers: Np = 2p−1 (2p − 1) where p and 2p − 1 are prime numbers. Euclid had listed the first four perfect numbers: 6; 28; 496; and 8128. A manuscript of 1456 gave the fifth perfect number: 33,550,336. Gradually mathematicians found further perfect numbers (which are very rare). In 1652 the Polish polymath Jan Brożek noted that there was no perfect number