I shall in the first part confer with you about your _Arithmetica Infinitorum_, and afterwards compare our manner of elocution; then your politics; and last of all your grammar and critics. Your spiral line is condemned by him whose authority you use to prove me a plagiary, (that is, a man that stealeth other men’s inventions, and arrogates them to himself), whether it be Roberval or not that writ that paper, I am not certain. But I think I shall be shortly; but whosoever it be, his authority will serve no less to show that your doctrine of the spiral line, from the fifth to the eighteenth proposition of your _Arithmetica Infinitorum_, is all false; and that the principal fault therein (if all faults be not principal in geometry, when they proceed from ignorance of the science) is the same that I objected to you in my _Lessons_. And for the author of that paper, when I am certain who it is, it will be then time enough to vindicate myself concerning that name of plagiary. And whereas he challenges the invention of your method delivered in your _Arithmetica Infinitorum_, to have been his before it was yours, I shall, I think, by and by say that which shall make him ashamed to own it; and those that writ those encomiastic epistles to you ashamed of the honour they meant to you. I pass therefore to the nineteenth proposition, which in Latin is this: your geometry! “_Si proponatur series quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum) continue crescentium, a puncto vel 0 inchoatarum, (puta ut 0. 1. 4. 9. 16. etc.), propositum sit, inquirere quam habeat illa rationem ad seriem totidem maximæ æqualium._ “_Fiat investigatio per modum inductionis ut_ (_in prop. 1_) _Eritque_, (0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6) (0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (1)/(3) + (1)/(12) (0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) = (1)/(3) + (1)/(18) _et sic deinceps_. “_Ratio proveniens est ubique major quam subtripla seu (1)/(3); excessus autem perpetuo decrescit prout numerus terminorum augetur (puta (1)/(6) (1)/(12) (1)/(18) (1)/(24) etc.) aucto nimirum fractionis denominatore sive consequente rationis in singulis locis numero senario (ut patet) ut sit rationis provenientis excessus supra subtriplam, ea quam habet unitas ad sextuplum numeri terminorum post 0; adeoque._” That is, if there be propounded a row of quantities in duplicate proportion of the quantities arithmetically proportional (or proceeding in the order of the square numbers) continually increasing; and beginning at a point or 0; let it be propounded to find what proportion the row hath; to as many quantities equal to the greatest; Let it be sought by induction (as in the first proposition).