Passing then to augmented series... and diminished... or altered... constituted from sums or differences of two or more other series. ...[I]t was not… - John Wallis

I made no Scruple of diverting (from the common Road of Studies then in fashion) to any part of Useful Learning. Presuming, that Knowledge is no Burthen; and, if of any part thereof I should afterwards have no occasion to make use, it would at least do me no hurt; And what of it l might or might not have occasion for, I could not then foresee.

At Christmass 1631, (a season of the year when Boys use to have a vacancy from School,) I was, for about a fortnight, at home with my Mother at Ashford. I there found that a younger Brother of mine (in Order to a Trade) had, for about 3 Months, been learning (as they call'd it) to Write and Cipher, or Cast account, (and he was a good proficient for that time,) When I had been there a few days; I was inquisitive to know what it was, they so called. And (to satisfie my curiosity) my Brother did (during the Remainder of my stay there before I return'd to School) shew me what he had been Learning in those 3 Months. Which was (besides the writing a fair hand) the Practical part of Common Arithmetick in Numeration, Addition, Substraction, Multiplication, Division, The Rule of Three (Direct and Inverse) the Rule of Fellowship (with and without, Time) the Pule of False-Position, Rules of Practise and Reduction of Coins, and some other little things. Which when he had shewed me by steps, in the same method that he had learned them; and I had wrought over all the Examples which he before had done in his book; I found no difficulty to understand it, and I was very well pleased with it: and thought it ten days or a fortnight well spent. This was my first insight into Mathematicks; and all the Teaching I had.

I... began... with simple series... of quantities in arithmetic proportion, or... their squares, cubes, etc. and then... their square roots, cube roots, etc. and powers composed of these... square roots of cubes etc. or... whatever... composites, whether the power was rational or... irrational. ...Whence a general theorem emerged... Proposition 64. But also... the quadrature... of the simple parabola... of all higher parabolas, and their complements, which no-one before... achieved. I... had enlarged geometry; for... there may now be taught by a single proposition the quadrature or all higher of infinitely many kinds... by one general method. ...I felt it would be welcome ...to the mathematical world ...also I saw ...the same doctrine widened ...I have related everything, whether conoids or pyramids, either erect or inclined, to cylinders and prisms. ...I saw ...as a direct consequence an almost completed teaching of spirals; and indeed I have taught the comparison with a circle... But also that teaching... was capable of extension...