That which deceived you was partly this, that you think, as you do in your _Elenchus_ , that these fractions (1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) &c. are proportions, as if (1)/(12) were the proportion of one to twelve, and consequently (2)/(12) double the proportion of one to twelve; which is as unintelligible as school-divinity; and I assure you, far from the meaning of Mr. Ougthred in the sixth chapter of his _Clavis Mathematica_, where he says that 4(3)/(7) is the proportion of 31 to 7; for his meaning is, that the proportion of 4(3)/(7) to one, is the proportion of 31 to 7; whereas if he meant as you do, then 8(6)/(7) should be double the proportion of 31 to 7. Partly also because you think (as in the end of the twentieth proposition) that if the proportion of the numerators of these fractions (1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) to their denominators decrease eternally, they shall so vanish at last as to leave the proportion of the sum of all the squares to the sum of the greatest so often taken, (that is, an infinite number of times), as one to three, or the sum of the greatest to the sum of the increasing squares, as three to one; for which there is no more reason than for four to one, or five to one, or any other such proportion. For if the proportions come eternally nearer and nearer to the subtriple, they must needs also come nearer and nearer to subquadruple; and you may as well conclude thence that the upper quantities shall be to the lower quantities as one to four, or as one to five, &c. as conclude they are as one to three. You can see without admonition, what effect this false ground of yours will produce in the whole structure of your _Arithmetica Infinitorum_; and how it makes all that you have said unto the end of your thirty-eighth proposition, undemonstrated, and much of it false. The thirty-ninth is this other lemma: “_In a series of quantities beginning with a point or cypher, and proceeding according to the series of the cubic numbers, as O. 1. 8. 27. 64, &c. to find the proportion of the sum of the cubes to the sum of the greatest cube, so many times taken as there be terms_.” And you conclude that “_they have a proportion of 1 to 4_;” which is false.