(^480) 16. THE CALCULUS
himself, his brother Jakob, and by both Newton and Leibniz.^6 According to Wood-
house (1810, p. 150), Newton's anonymously submitted solution was so concise and
elegant that Johann Bernoulli knew immediately who it must be from. He wrote,
"Even though the author, from excessive modesty, does not give his name, we can
nevertheless tell certainly by a number of signs that it is the famous Newton; and
even if these signs were not present, seeing a small sample would suffice to recognize
him, as ex ungue Leonem."^7
Euler. Variational problems were categorized and systematized by Euler in a large
treatise in 1744 named Methodus inveniendi lineas curvas (A Method of Finding
Curves). In this treatise Euler set forth a series of problems of increasing com-
plexity, each involving the finding of a curve having certain extremal properties,
such as minimal length among all curves joining two points on a given surface.^8
Proposition 3 in Chapter 2, for example, asks for the minimum value of an integral
/ Zdx, where Æ is a function of variables, x, y, and ñ = y^1 = ^. Based on his
previous examples, Euler derived the differential equation
0 = Ndx-dP,
where dZ = Ì dx -f- Í dy + Ñ dp is the differential of the integrand Z. Since
Í = ff an(^ Ñ ~ ' tn's eQuation could be written in the form that is now the
basic equation of the calculus of variations, and is known as Euler's equation:
<)Z d_/dZ\
dy dx í dy')
In Chapter 3, Euler generalized this result by allowing Æ to depend on addi-
tional parameters and applied his result to find minimal surfaces. In an appendix
he studied elastic curves and surfaces, including the problem of the vibrating mem-
brane. This work was being done at the very time when Euler's friend Daniel
Bernoulli was studying the simpler problem of the vibrating string. In a second
appendix he showed how to derive the equations of mechanics from variational
principles, thus providing a unifying mathematical principle that applied to both
optics (Fermat's principle) and mechanics.
Lagrange. The calculus of variations acquired "variations" and its name as the
result of a letter written by Lagrange to Euler in 1755. In that letter, Lagrange
generalized Leibniz' differentials from points to curves, using the Greek ä instead of
the Latin d to denote them. Thus, if y — f(x) was a curve, its variation 6y was a
small perturbation of it. Just as dy was a small change in the value of y at a point, <5y
was a small change in all the values of y at all points. The variation operator ä can
be manipulated quite easily, since it commutes with differentiation and integration:
ä÷/ = (äõ)' and ä j Æ dx = J äÆ dx. With this operator, Euler's equation and its
many applications, were easy to derive. Euler immediately recognized the usefulness
of what Lagrange had done and gave the new theory the name it has borne ever
since: calculus of variations.
Lagrange also considered extremal problems with constraint and introduced
the famous Lagrange multipliers as a way of turning these relative (constrained)
(^6) Newton apparently recognized structural similarities between this problem and his own optimal-
streamlining problem (see Goldstine, 1980, pp. 7-35).
(^7) A Latin proverb much in vogue at the time. It means literally "from [just] the claw [one can
recognize] the Lion."
(^8) This problem was Example 4 in Chapter 4 of the treatise.