PREFACE

459,818,240; which, when multiplied by 3, produces the number 1,379,454,720. Its aliquot parts These are proper divisors of a number, which are integers that divide the number exactly, excluding the number itself. are triple the original number. The following numbers 30,240, 32,760, and 23,569,920 are of the same kind, along with infinite others. Concerning these, one may consult our Harmonie Universelle original: "Harmonia nostra". This is Mersenne's 1636 work on music and mathematics., in which 14,182,439,040 and others are found to be subquadruple This means the sum of the number's divisors is four times the number itself. to their aliquot parts. They are also in a given ratio with their aliquot parts.

There are also other numbers which they call amicable original: "amicabiles". These are pairs of numbers where the sum of the divisors of one equals the other number.. They are called this because they are mutually restored by their aliquot parts. Examples of these are the smallest ones, 220 and 284. The parts of the latter produce the former; in turn, the parts of the former perfectly restore the latter. You will also find 18,416 and 17,296; likewise 9,437,036 and 4,363,584; and countless others.

At this point, it is worthwhile to note that the twenty eight numbers presented as perfect numbers A perfect number is a positive integer that is equal to the sum of its proper divisors, such as 6 or 28. by Pietro Bongo original: "Petro Bungo". Bongo was a 16th century author who wrote on the mystical properties of numbers. in chapter 28 of his book On Numbers are not all perfect. Indeed, twenty of them are imperfect. Thus, he only has eight perfect numbers: 6, 28, 496, 8,128, 33,550,336, 8,589,869,056, 137,438,691,328, and 2,305,843,008,139,952,128. These correspond to positions 1, 2, 3, 4, 8, 10, 12, and 29 in Bongo's table. Only these are perfect. Those who own Bongo’s work should apply this remedy to his error.

Furthermore, perfect numbers are so rare that only eleven have been found so far. That is, three others different from Bongo's list. There is no other perfect number among those eight unless you exceed the exponent number 62 of the double progression starting from 1. The ninth perfect number is the power of exponent 68 minus 1. The tenth is the power of exponent 128 minus 1. Finally, the eleventh is the power of 258 minus 1; that is, the power of 257, shortened by one unit, multiplied by the power of 256. Mersenne is describing what are now called Mersenne Primes, numbers of the form 2^p - 1. A number is perfect if it is of the form (2^(p-1))(2^p - 1) where 2^p - 1 is prime.

Whoever finds another eleven should know that they have surpassed all analysis that has existed until now. Meanwhile, let them remember that there is no perfect number from the power of 17,000 to 32,000. No interval of powers can be assigned so large that it is not devoid of perfect numbers. For example, if the exponent were 950,000, there would be no number in the double progression up to 2,090,000 that serves for perfect numbers; that is, one that, being one less than a unit, exists as a prime.

From this it is clear how rare perfect numbers are. They are rightly compared to perfect men. It is one of the greatest difficulties in all of Mathematics original: "Matheseos". to exhibit the prescribed multitude of perfect numbers. This is similar to the difficulty of recognizing whether given numbers consisting of 15 or 20 characters are prime or not. Even an entire century would not suffice for this examination by any method known so far.

XX. In the books on Hydrostatics and Navigation original: "Hiftrodromiæ"., many things are supplied that were missing in the Hydraulics. Mutual light will shine from these. We also propose a ship which travels under the waters of the Ocean, or any others...