I had once added to these nineteen articles a twentieth, which was this: “_If from a point in the circumference there be drawn a cord, and a tangent equal to it, the angle which they make shall be double to the aggregate of all the angles made by the cords of all the equal arches into which the arch given can possibly be divided_.” Which proposition is true, and I did when I writ it think I might have use of it. But be it, or the demonstration of it true or false, seeing it was not published by me, it is somewhat barbarous to charge me with the faults thereof. No doctor of humanity but would have thought it a poor and wretched malice, publicly to examine and censure papers of geometry never published, by what means soever they came into his hands. I must confess that in these words, _in such kind of progression arithmetical_ (that is, which begins with 0) _the sum of all the numbers taken together, is equal to half the number that is made by multiplying the greatest into the least_, there is a great error; for by this account these numbers, 0, 1, 2, 3, 4, taken together, should be equal to nothing. I should have said they are equal to that number which is made by multiplying half the greatest into the number of the terms. There was therefore, if those words were mine (for truly I have no copy of them, nor have had since the book was printed, and I have no great reason, as any man may see, to trust your faith) a great error in the writing, but not an erroneous opinion in the writer. The demonstration so corrected is true. And the angles that have the proportions of the numbers 1, 2, 3, 4, are in the table of your _Elenchus_ , fig. 12, the angles G A D, H D E, I E F, K F B. And if the divisions were infinite, so that the first were not to be reckoned but as a cypher, the angle C A B would be double to them altogether. This mistake of mine, and the finding that I had made no use of it in the whole book, was the cause why I thought fit to leave it quite out. But your professorships, could not forbear to take occasion thereby, to commend your zeal against _Leviathan_ to your doctorships of divinity, by censuring it. ========== OF THE FAULTS THAT OCCUR IN DEMONSTRATION. TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN THE UNIVERSITY OF OXFORD. LESSON V.