Understanding the‘ScientificRevolution’ 155Appendix A
(From Oresme,Quaestiones super Euclidem)Next we inquire whether an addition to any magnitude could be made by proportional parts.
First, it is argued that it cannot be, since then it would follow that a magnitude could be
capable of being increased to an actual infinity. This consequence contradicts what Aristotle says
in the third book ofPhysicaand also Campanus’s statement [in his commentary on the Common
Notions in Euclid I], where he distinguishes between a number and a magnitude, in that a number
can increase indefinitely and not decrease indefinitely, but the reverse is true of a magnitude.
Proof of the consequence: From the fact that the addition takes place indefinitely it follows
that the increase too takes place indefinitely.
Against this it is argued: anything that is takenawayfrom one magnitude can be added to
another. It is possible to takeawayfrom a magnitude an infinite number of proportional parts,^17
therefore it is also possible to prove that it can be increased by an infinite number of parts.
[This is the only ‘argument against’ the possibility of addition, and follows fairly closely the
quaestiostructure with its objection, reference to authority, and so on. Notice (and this agrees with
the Duhem thesis, perhaps) that Aristotle’s authority is cited in argument, but is not treated as
conclusive—in fact, the counter-arguments override it. After this, Oresme goes in more detail into
the mathematics at issue, beginning with some definitions of types of ratio, which I omit.]
Secondly, it must be noted that if an addition were made to infinity by proportional parts in a
ratio of equality or of greater inequality, the whole would become infinite; if, however, this addition
should be made [by proportional parts] in a ratio of lesser inequality, the whole would never become
infinite, even if the addition continued into infinity. As will be declared afterward, the reason is
because the whole will bear a certain finite ratio to the first [magnitude] assumed to which the
addition is made...[Here follow some definitions on fractions.]
The first proposition is that if a one-foot quantity should be assumed and an addition were made
to it into infinity according to a subdouble [that is one-half] proportion so that one-half of one
foot is added to it, then one-fourth, then one-eighth, and so on into infinity by halving the halves
[lit. doubling the halves], the whole will be exactly twice the first [magnitude] assumed. This is clear,
because if from something one takesawaysuccessively these parts, then [one is left with nothing,
and so] from the double quantity one has takenaway thedouble, as appears by question 1 [which
was about subtraction]; and so by a similar reasoning, if they are added.
The second proposition is this, that if a quantity, such as one foot, were assumed, then a
third were added and then after a third [of that] and so into infinity, the whole would be precisely
one foot and a half, or in the sesquialterate proportion [this is the medieval terminology for the
ratio of 3 to 2]. Furthermore, this rule should be known: We must see how much the second part
falls short of the first part, and how much the third falls short of the second, and so on with the
others, and denominate this by its denomination, and then the ratio of the whole aggregate to the
quantity [first] assumed will be just as a denominator to a numerator. [This looks very obscure, but
the meaning seems to be that, if your ratio is a fractionqso the series isa+aq+aq^2 +···, then
the ‘falling short’ is 1−q; and you invert this—exchange the denominator and the numerator—
to get the sum 1/( 1 −q), which is the ratio of the sum to the first terma. There is no proof.]
- This is the substance of the previous ‘question’, which deals with the successive subtractions which exhaust a magnitude, as
in Euclid X.1.