Key dates in the development of calculus notation
Derivative
1675 - The symbols dx, dy, and dx/dy were introduced by Leibniz.
1797 - The symbols f'(x) for the first derivative, f''(x) for the second derivative, etc., were
introduced by Lagrange.
Partial derivative
1770 - The "curly d" was first used by Caritat. Throughout his paper, both dz & z either
denoted two partial differences of z, where one of them was with respect to x, and the
other, with respect to y, or dz and z were employed as symbols of total differential,
and of partial difference, respectively.
1786 - The "curly d" was first used in the form
by Legendre.
1841 - Jacobi re-introduced the symbol after Legendre abandoned it. The "curly d" symbol is
sometimes called the "rounded d" or "curved d" or Jacobi’s delta. It corresponds to
the cursive "dey" (equivalent to our d) in the Cyrillic alphabet.
Integral
1675 - The integral symbol was first used by Leibniz. The integral symbol was actually a
long letter S for “summa.” Before introducing the integral symbol, he wrote omn. For
“omnia” in front of the term to be integrated.
1675 - Leibniz first placed dx after the integral symbol, replacing
.
1704 - Newton wrote a small vertical bar above x to indicate the integral of x. He wrote two
side-by-side vertical bars over x to indicate the integral of (x with a single bar over it).
Another notation he used was to enclose the term in a rectangle to indicate it’s
integral. Newton’s symbolism for integration was considered defective because the x
with a bar could be misinterpreted as x-prime and the placement of a rectangle about
the term was difficult to print, therefore Newton’s symbolism was never popular.
Limits of integration
1768 - Euler was the first to use a symbol to notate limits of integration. He wrote the limits
in brackets and used the Latin words ab and ad. Prior to this limits of integration were
indicated only in words
1822 - The modern definite integral symbol
was originated by Fourier.
Integration around a closed path
1917 - Sommerfeld first used the integral symbol with a circle in the middle ∮ to denote the
integral round a closed path.
Limit
1786 - lim. (with a period) was used first by L’Huilier.
The arrow notation for limits
1850 - Weierstrass began to use
1905 - Our present day expression
.
originated by Leathem.
Infinity
1655 - The infinity symbol
was introduced by Wallis.
Delta to indicate a small quantity
1706 - Bernoulli first used δ to denote the difference of functions.
Epsilon
1821 - Cauchy first used epsilon ε to denote “a number as small as one may wish”.
The vector differential operator
1837 - now written as
and called nabla or del, was introduced by Hamilton.