will be to the sum of the Progression from 1, ad Infinitum, as 3 to 2−¼−⅛−〈 math 〉〈 math 〉, &c. that is, by Consect. 2 and 3, as 3 to 2−½, that is, as 3 to 1 ½, or as 2 to 1. Q.E.D.
X. The Sum of any Duplicate Arithmetical Progression (i. e. a Progression of Squares of whole numbers ascending) continued from 1 ad Infinitum, is subtriple of the Sum of as many Terms equal to the greatest as is the number of Terms: For any such finite Progression is greater than the subtriple Proportion, but approaches nearer and nearer to it continually, by how much the farther the Series of the Progression is carried on. Thus the Sum of 3 Terms 1, 4, 9=14 is to thrice 9=27 as 1 〈 math 〉〈 math 〉, or 1 〈 math 〉〈 math 〉, or 1+½+〈 math 〉〈 math 〉 to 3 (dividing both sides by 9,) the Sum of six Terms, 1, 4, 9, 16, 25, 36, viz. 91. to six times 36, i. e. to 216 (dividing both sides by 72) is as 1+¼+〈 math 〉〈 math 〉 to 3; and the Sum of 12 Terms 650, to 12 times 144, i. e. to 1728 (dividing both sides by 576) is as 1+⅛+〈 math 〉〈 math 〉 to 3, &c. the Fractions adhering to them thus constantly decrea∣sing, some by their half parts, others by three quarters (for 〈 math 〉〈 math 〉 is 〈 math 〉〈 math 〉; therefore the first decrement is 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉, is 〈 math 〉〈 math 〉; there∣fore the second decrement is 〈 math 〉〈 math 〉, &c.) Wherefore the Sum of the Infinite Progression will be to the Sum of the like number of Terms equal to the greatest, as 〈 math 〉〈 math 〉, &c. to 3, that is, by Consect. 3 and 8, as 1 to 3. Q.E.D.
XI. The Sum of a triplicate Arithmetical Progression (i. e. ascending by the Cubes of the Cardinal Numbers) proceeding from 1 thro' 27, 64, &c. ad Infinitum, is Subquadruple of ••he Sum of the like number of Terms equal to the greatest. For the Sum of 4 Terms, 1, 8, 27, 64, i. 100, to 4 times 64, i. e. 256 (dividing both sides by 64) will be found to be as 1+½+〈 math 〉〈 math 〉 to 4; but the Sum of 8 Terms, 1, 8, 27, 64, 125, 216, 343, 582, i. e. 1296 to 8 times 512, that is, 4096 (dividing both Sides by 1024.) will be found to be as 1+¼+〈 math 〉〈 math 〉 to 4, &c. The adhering Fractions thus