According to Descartes

Objectivity. As Descartes' Discourse on Method instructed, Leeuwenhoek was:

careful to avoid precipitancy and prejudice, and to comprise nothing more in [his] judgment than what was presented to [his] mind so clearly and distinctly as to exclude all ground of doubt.

Repeatedly in his letters, Leeuwenhoek calls his speculations what they are and keeps them separate from his observations. The little animals were as clear, Leeuwenhoek wrote, as "sandgrains that one might bestrew upon a piece of black taffety silk".

Analysis. Descartes' second precept was:

to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.

Leeuwenhoek walked the readers through his procedure of dividing a thin glass pipe into sections and showed them his arithmetic.

This small quantity of Water I gather up into a very slender glass-pipe, dividing by this means that little water into 25 or 30 parts, of which I observe one part after another.

He concluded:

... and therefore 1000000 living Creatures in one drop of water. In which computation I rather lessen than heighten the number. ...

Yet I need not yield, that I ever do exaggerate my numbers.

Quantification. Descartes' final precept was:

And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

Leeuwenhoek not only enumerated in the sense of counting. He sent drawings with almost all of his letters, and referred to them in the text. The members of the Royal Society could do the math, as much as it defied common sense.