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This comment cites Wikipedia in stating that, before the development of the modern-day subject of calculus (i.e. analysis), the term "calculus" referred to general mathematics.
Who first used the term "calculus" in mathematics, and what specifically as it applied to? I do know that the Greeks used the word "mathematics" in ancient times to refer to what we would today call "mathematics", and the word took on different meanings at different points in time.
According to Carl B. Boyer, "The history of the calculus and its conceptual development", Dover Publications 1959, page 98,
The improved notation led also to methods which were so much more facile in application than the cumbrous geometrical procedures of Archimedes, of which they were modifications, that these methods were eventually recognized as forming a new analysis—the calculus. The period during which this transformation took place may be considered as the century preceding the work of Newton and Leibniz.
(The question is complicated by the fact that most mathematicians were writing in Latin.) If you are asking when the word "calculus" was used to refer to the differential and/or integral calculus, that must surely post-date the development of those subjects. For example, Richard Suiseth was known as Calculator in the 14th century, and experts on what we now call arithmetic were called "reckoners" in the middle ages. The word "reckon" is really the German word "Rechen", which means "calculate". So "Calculator" is just Latin for "Rechner". So the word "calculus" in one form or another was around for a while in regard to arithmetic. The question is somewhat confused by the fact that what we call "mathematics" was actually called "geometry" until only a few centuries ago, and mathematicians were called "geometers", even into the 19th century. The term "mathematics" in Greek refers to knowledge in general.
Perhaps the better question to ask is when were the terms "differential calculus" and "integral calculus" first used. The guy who taught Newton his basic calculus (before Newton developed it further) was Isaac Barrow, who wrote a book "Geometrical Lectures", showing that this subject was still considered to be part of geometry at that time. According to Boyer, page 190, Newton used the word "analysis" for his first publication on calculus.
The first notice of his calculus was given, however, in 1669, in De analysi per aequationes numero terminorum infinitas.
And integration was originally called "quadrature", as in Newton's De quadratura. (Boyer, page 205.) It seems to me that it was Leibniz who first promoted the term "calculus" in the modern sense. He wrote a book Historia et origo calculi differentialis a year or two before his death. (Boyer, page 215.) In 1712, the Royal Society committee reported that
The differential method is one and the same with the method of fluxions, excepting the name and mode of notation [...]
(Boyer, page 221.) This shows that Newton was not even using the term "differential calculus". (The term "differential method" referred to Leibniz's method.)
According to Boyer, page 67, the word "integral" originated with Leibniz also, but on the suggestion of the Bernoulli brothers. So it seems to me that the term "calculus", with that name, developed only during the late 17th century, more in Europe than in Britain, where the term was resisted for some time.
PS. If you just want to know when the word "calculus" was first used for calculating (as opposed to the meaning of small stones or pebbles), my Latin dictionary (John T. White, 1902, 1923) gives quotes from Cicero and Livius where the word was used in the sense of "reckoning, computing, calculating". That puts it back to 2000 years ago, plus or minus about 30 years.
PPS. Perhaps I could also mention that what was called "calculus" in ancient Roman times and up until the Renaissance in Europe was not considered to be part of the same subject as geometry, which was taught at the universities and the precursors of universities. There was a sharp divide between the advanced mathematics of Euclid, Apollonious and Archimedes and the market-place arithmetic which is well-documented since the beginning of writing 5500 years ago. Everyone in these 5.5 millennia would have learned the basic marketplace arithmetic as part of growing up. It was not a study as such. It did not need a university education. Nor was it generally learned from books. So to call that marketplace "calculus" or "reckoning" part of mathematics is not really correct. If you ask a mathematician right now to add up a supermarket invoice with 30 items, they might have some difficulties, but we would not think that this indicates a lack of mathematical ability. That's just the "reckoning" of the marketplace, and carpentry, book-keeping, and dozens of other trades. So in my opinion, "calculus" was not used for a really mathematical subject until the 16th or 17th century. Until then, it was only applied to marketplace addition, subtraction, multiplication and division. Calling that mathematics would be like calling a road-sign literature.
Your question was: Who was the first person to use the term “calculus”? I cannot say who the first individual was, but in any case he or she was an ancient Roman: the Latin word “calculus” means “small stone, pebble”, then “a stone used in reckoning on the counting-board” and then “reckoning, computing, calculating”. This usage is known from the time of Cicero (1st century BC) onwards. The word “calculus” itself comes from “calx” “limestone” (as in “calcium”), which is probably a borrowing from Greek “chalix”, with the same meaning.
You might want to look here: http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dcalculus
As per fdb's answer, the latin word calculus was used with the arithmetical meaning early on.
See Anthony Lo Bello, Origins of mathematical words : A Comprehensive Dictionary of Latin, Greek, and Arabic Roots (2013), page 50 :
calculate The Latin word for stone is calx, calcis. The addition of the ending -ulus to the stem produces the diminutive calculus, which means a small stone or pebble. As a medical term, it is used of bladder, gall, and kidney stones, and even the gritty accumulation on the teeth. Since such pebbles were used as counters in counting, the verb calculo, calculare, calculavi, calculatus came into existence with the meaning to count.
calculus This is the Latin word for a small stone. It came to be used by the late seventeenth-century mathematicians as a technical term for any theory that laid the foundations of a general method to calculate the solutions of certain types of problems and then, to those theories that solved the problems of tangent lines (differential calculus) and quadrature (integral calculus).
But the use of calculus in mathematical books is not easy to trace.
We have take into account :
The Compendious Book on Calculation by Completion and Balancing (Arabic: Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala), written by Muḥammad ibn Mūsā al-Khwārizmī around 820 and translated into Latin by Robert of Chester in the mid-12th century as Liber algebrae et almucabola.
Leonardo Fibonacci or Leonardo Pisano, Liber Abaci, composed in 1202; we can translate it as "The Book of Calculation".
Richard Swineshead, Liber calculationum ("Book of Calculations"), written around 1350.
During the Renaissance, we have a tension between re-discovery of ancient Greek mathematics and new discoveries. Thus, we have a flourishing of new names for rediscovered or new-founded disciplines.
The first one is algebra; we can list :
Girolamo Cardano, with his Artis magnae, sive de regulis algebraicis of 1545 ("The Great Art") dedicated to the new algebra.
Rafael Bombelli, with his 1572 book, L'Algebra.
François Viète, with his In Artem Analyticien Isagoge (Introduction to the art of analysis) (1591).
In Viète, Analysis is not yet the modern discipline; for its "birth", we have to see Newton's 1669 manuscript : De analysi per aequationes numero terminorum infinitas (published in 1704).
The first usage of calculus with the new meaning is due to Leibniz; see G.W.Leibniz (editor : J.M.Child), The Early Mathematical Manuscripts (1920 - also Dover reprint) :
Analysis Tetragonistica Ex Centrobarycis (Analytical quadrature by means of centers of gravity), dated 1675 (page 65)
Methodi tangentium directae compendium calculi ... (Compendium of the calculus of the direct method of tangents ...), dated 1675, page 111
Calculus Tangentium differentialis(Differential calculus of tangents), dated 1676, page 124
Elementa Calculi Novi pro differentiis et summis, tangentibus et quadraturis, maximis et minimis, dimensionibus linearum, superficierum, solidorum, allisque communem calculum transcendentibus (The Elements of a New Calculus for Differences and Sums, Tangents and Quadratures, maxima and minima, the measurement of lines, surfaces and solids, and other things which transcend the usual sort of calculus); the manuscript is undated, but appears to have been compiled sometime later than 1677 and prior to 1680, page 136.
See the Commercium Epistolicum for a variety of terms : Analysis, Newton's Method of Series, Method of Fluxions, Dr.Barrow [and] Mr.Leibnitz : their Methods of Tangents, Leibniz's calculo differentiali.
Presumably, the success of Analysis is due to :
I'm surprised no one has mentioned the story which I have commonly heard. Yes "calculus" means pebble or stone in Latin. But just saying this...
Calculus has always had, and still has, a more general meaning than the differentiation and integration. It literally means a method of calculating, and the etymology goes back to small stones used as counters
... misses a more interesting story. A "calculus," or calculi, essentially enabled Roman taxi cabs to figure a fare.
The odometer of Vitruvius was based on chariot wheels of 4 feet (1.2 m) diameter turning 400 times in one Roman mile (about 1400 m). For each revolution a pin on the axle engaged a 400 tooth cogwheel thus turning it one complete revolution per mile. This engaged another gear with holes along the circumference, where pebbles (calculus) were located, that were to drop one by one into a box. The distance traveled would thus be given simply by counting the number of pebbles. Whether this instrument was ever built at the time is disputed. Leonardo da Vinci later tried to build it himself according to the description, but failed. However, in 1981 engineer Andre Sleewyck built his own replica, replacing the square-toothed gear designs of da Vinci with the triangular, pointed teeth found in the Antikythera mechanism. With this modification, the Vitruvius odometer functioned perfectly.
