"This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles…. for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals." (John Wallis, "Arithmetica Infinitorum", 1656)
"The same thing is confirmed through the previous proposition since, for example, an infinite collection from which units can be subtracted (not only ten but infinitely many) while the collection remains infinite is, obviously, infinitely greater before the subtraction takes place than after; thus, since the collection does not cease being infinite after the subtraction, the infinite will be, as such, infinitely smaller than it was earlier. You could say that this is in conflict with the generally accepted thesis that holds that the terms 'greater' and 'smaller' can only apply to finite quantities but not to infinite quantities; or, at least, that they can be applied to infinite quantities only in a very improper way. I reply that this idea has the following feature, namely that it is widespread. This fact notwithstanding, I would say, with permission, that it also has this other feature, namely that its ground is nothing else but a false notion of infinity. Moreover, the advantage it offers, which consists in apparently solving some difficulties that are usually put forth by denying that 'greater' and 'smaller' are properties that can be predicated of infinity, does not subsist for it ends up not resolving the difficulties." (Emmanuel Maignan, 1673)
"Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum's composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysics except he who has passed through this [labyrinth]." (Gottfried W Leibniz, "Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae", 1676)
"Where, by the way, we may observe a great difference between the proportion of Infinite to Finite, and, of Finite to Nothing. For 1/∞, that which is a part infinitely small, may, by infinite Multiplication, equal the whole: But 0/1 , that which is Nothing can by no Multiplication become equal to Something." (John Wallis, "Treatise of Algebra", 1685)
"A quantity diminished or enlarged by an infinitely smaller quantity is neither diminished nor enlarged." (Johann Bernoulli, cca 1691-92)
"Since the nature of differentials […] consists in their being infinitely small and infinitely changeable up to zero, in being only quantitates evanescentes, evanescentia divisibilia, they will be always smaller than any given quantity whatsoever. In fact, some difference which one can assign between two magnitudes which only differ by a differential, the continuous and imperceptible variability of that infinitely small differential, even at the very point of becoming zero, always allows one to find a quantity less than the proposed difference." (Johann Bernoulli, cca. 1692–1702)
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