12 are further defined as proportional ( सजातीय), when the second is the same multiple of bhe first as the fourth is of the third. Like plane numbers ( सजातीयक्षेत्रफळ ) and like cube numbers ( सजातीयधनफछ) are those which have bheir sides proportional. Finally a perfect number ( पूर्णा) is one which is equal to the sum of all its aliquot parts, as 6.* The Seventh Book demonstra tes in general the most common properties of numbers, chiefly of prime and composite numbers, and partly treats of the compa rison of one number with another. The enunciations of a few propositions will make this clear.+ 'If of two numbers the less is continually taken from the greater until unity is left, the two numbers are incommensurable or prime to one another.' To find the greatest common measure of two or more quantities. small quantity is a part of a large quantity or of its multiple. If two quantities be the same part of two other quantities, the sum of the first two shall be the same part of the sum of the other two.' + If from two number two other numbers in the same radio be taken, the remainders shall be in bhe same ratio. The product of the multiplicand by the multiplier is bhe same as bhat of the multiplier by the multiplicand.If bhere are small numbers in a certain ratio, sueh that smaller numbers in the same ratio cannot be found, then these numbers shall be prime to one another' * If a certain number is prime to another, its square also shall be prime to it' + If two numbers are incommensurable, their squares as well as bheir cubes shall also be incommensurable. < To find the least common multiple of two or more numbers.’ ‘To find the least common multiple which can be measured by many factions'+ The book contains 89 propositionsThe plane and solid numbers, their sides and proportion, the properties of square and cube numbers, the natures and conditions of their sides, and the mean proportional numbers of plane, solid, square, and cube numbers form mainly the subject of bhe
- The aliquot parts oft 6 are 1, 2 and 3 and these together make up the
number, 6. The numbers to whion this properdy belongs are 6;28;B6;8128; 33,560,336; 8,589,869,056; 187488,691,3928; and 2805848,008,189,952,123. All pertect numbers terminate with a or 28. Vide Chambers' 'Popular Educator.' + Props, 1, 2 and 3, 4, 5, 7, 16, 21, 25, 2, 34 and 86, and 88 respectively, Book VII.
