TheCalculus 175
we have claiming pride in the discovery, such as the one which opens this section, come in the
main from some time later when others had taken the trouble to understand and explain it. And
we can see the prestige of the calculus growing with the priority quarrel; that something had
been plagiarized meant that it was worth plagiarizing, and vice versa. While the paper was clearly
found more than difficult at the time, some of its faults cannot be judged by later standards. Since
Descartes at least, the importance of proof of results had been declining. As Guicciardini (citing
Bos as authority) says:
[I]t was common in the early [and one might add, late] seventeenth century to give priority and publicity to the
geometrical construction which solves the problem, rather that to the analysis necessary to achieve such a construction
(an analysis which was often kept hidden by mathematicians)...(Guicciardini 1999, p. 98)This is not to say that construction and proof were not often argued about, and unproved results
or inadequate proofs challenged. However, Leibniz’s paper was too murky even to warrant such a
challenge, and only a series of later explainers (and a stream of errata issued in the journal) finally
made it readable to later generations.
Interestingly, almost as an afterthought to the confusions of the paper itself, there appears a hint
of what the new methods can do:
It is required to find a curveWWsuch that, its tangentWCbeing drawn to the axis,XCis always equal to a given
constant linea. [Fig. 4.] ThenXWorwis toXCoraasdwis todx.Ifdx(which can be chosen arbitrarily) is taken
constant, hence always equal to, say,b, that is,xorAXincreases uniformly, thenw=abdw. Those ordinatesware
therefore proportional to theirdw, their increments or differences, and this means that if thexform an arithmetic
progression, then thewform a geometric progression. In other words, if theware numbers, thexwill be logarithms,
so that the curve is logarithmic.Leibniz claims this is a problem which Debeaune proposed to Descartes, and which Descartes could
not solve. Typically, this is not quite true; the problem is a simplified reformulation of Debeaune’s
problem, and Descartes did give a solution (see Fauvel and Gray 13.A.2 for the details, including–0.5 0 C 0.5 (^1) X 1.5 2 2.5
2
w
W
W
dw
dx
–1.5
4
6
8
–1
Fig. 4The example (exponential/logarithmic curve) from Leibniz’ paper.