In Herbert Goldstein's Classical Mechanics book, p. 75 (2nd ed.) it says:

Equations 3-18 and 3-20 are the two remaining integrations, and formally the problem has been reduced to quadratures...

It is not clear to me what a quadrature is. Is it simply the integration process through some numerical analysis method such as trapezoidal rule? Or is it the process of reducing the order of the initial partial differential equation by using constants of motion?

• Quadrature : In mathematics, quadrature is a historical term which means determining area. This term is still used nowadays in the context of differential equations, where "solving an equation by quadrature" means expressing its solution in terms of integrals. Commented Sep 19, 2018 at 18:47

So for example, $$\int_a^by(x) \ \mathrm{d}x \approx \sum_{n=1}^{N}\,y\left(a+\frac n N (b-a)\right)\frac{b-a}N, \tag{a quadrature}$$ $$\int_0^{2\pi}\sin^2(\theta) \ \mathrm{d}\theta = \pi, \tag{an exact integral}$$ and $$\int \sin^2(\theta) \ \mathrm{d}\theta = \frac \theta 2 - \frac 1 4 \sin(2\theta). \tag{an indefinite integral}$$