Why "integralis" over "summatorius"? It is written that Johann Bernoulli suggested to Leibniz that he (Leibniz) change the name of his calculus from "calculus summatorius" to "calculus integralis", but I cannot find their correspondence wherein Bernoulli explains why he thinks "integralis" is preferable to "summatorius." Can someone enlighten me? Thank you. 
 A: Bernoulli writes this to Leibniz in 20/30 April 1695. His justification for this choice of words is that "differentials are part of a whole" ("integer" being Latin for whole).
Leibniz replies (6/16 May 1695) that this appellation does not displease him, but that he considers his preferred term "sum" more illuminating and closer to the source of the notion.
See the Akademie-Ausgabe, III6A, pp. 348, 356:
http://www.leibniz-edition.de/Baende/ReiheIII.htm
A: If you wanted the correspondence. Maybe, this would be sure, the initial quotation states the exact point. As for the reason, some say that it was to rival Newton on proving that he had invented calculus first. The Bernoulli's too were involved in the controversy.This page explains the controversy.
A: I have no exact answer to your question, but I can provide you with some background. In late medieval Italian the adjective "integrale" was already used; its meaning is "total, entire". It is derived from the Latin word "integralis", which belongs to non-classical Latin (6th century A.C., so definitly non-classical): it appears in some comments to Cicerone's works. The Latin word "integer" is classical, instead. It has the same root of "tangere" (to touch) and "tangente" (yes, tangent!) and it means..."integer" or "pure/untouched".
Probably J. Bernoulli used the word "integralis" in 1690 to specify the property of "totality" or "completeness" of the operation, as the integral computes the "whole" area under a given curve.
Euler uses "integralis" in his "Institutiones calculi integralis" as an adjective.
It is funny to note that in the 18th century Luigi Guido Grandi and Maria Gaetana Agnesi discussed about "calcolo integrale" (i.e. Integral Calculus) using the word "integrale" as an adjective, while Riccati uses "integrale" as a noun.
