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From SALAS/HILLE
CALCULUS, 6th Edition
One and Several Variables, 1990
To a Roman in the days of the empire a "calculus" was a pebble used in counting and in gambling. Centuries later "calculare" came to mean "to compute", "to reckon", "to figure out". To the mathematician, physical scientist, and social scientist of today calculus is elementary mathematics (algebra, geometer, trigonometry) enhanced by the limit process.
Calculus takes ideas from elementary mathematics and extends them to a more general situation.
The origins can be traced back to ancient Greece. The ancient Greeks raised many questions (often paradoxical) about tangents, motion, area, the infinitely samll, the infinitely large questions that today are clarified and answered by calculus. Here and there the Greeks themselves provided answers (some very elegant), but mostly they provided only questions.
After the Greeks, progress was slow. Communication was limited, and each scholar was obliged to start almost from scratch. Over the centuries some ingenious solutions to calculus-type problems were devised, but no general techniques were put forth. Progress was impeded by the lack of a convenient notation. Algebra, founded in the ninth century by Arab scholars, was not fully systemized until the sixteenth century. Then, in the seventeenth century, Descartes established analystic geometry, and the stage was set.
The actual invention of calculus is credited to Sir Isaac Newton (1642 - 1727) and to Gottfried Wilhelm Leibniz (1646 - 1716), and Englishman and a German. Newton's invention is one of the few good turns that the great plague did mankind. The plague forced the closing of Cambridge University in 1665 and young Isaac Newton of Trinity College returned to his home in Lincolnshire for eighteen months of meditation, out of which grew his method of fluxions, his theory of gravitation, and his theory of light.
The method of fluxions is what concerns us here. A treatise with this title was written by Newton in 1672, but it remained unpublished until 1736, nine years after his death. The new method (calculus to us) was first announced in 1687, but in vague general terms without symbolism, formulas, or applications. Newton himself seemed reluctant to publish anything tangible about his new method, and it is not surprising that the development on the Continent, in spite of a late start, soon overtook Newton and went beyond him.
Leibniz started his work in 1673, eight years after Newton. In 1675 he initiated the basic modern notation : dx and integral sign. His first publications appeared in 1684 and 1686. These made little stir in Germany, but the two brothers Bernoulli of Basel (Switzerland) took up the ideas and added profusely to them. From 1690 onward calculus grew rapidly and reached roughly its present state in about a hundred years. Certain theoretical subtleties were not fully resolved until the twentieth century.
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Practical x 2 
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