  In the late 19th century, Swedish mathematician Lars Edvard Phragm\'{e}n
proposed a load-balancing approach for selecting committees based on approval
ballots. We consider three committee voting rules resulting from this approach:
two optimization variants -- one minimizing the maximal load and one minimizing
the variance of loads -- and a sequential variant. We study Phragm\'{e}n's
methods from an axiomatic point of view, focusing on properties capturing
proportional representation. We show that the sequential variant satisfies
proportional justified representation, which is a rare property for committee
monotonic methods. Moreover, we show that the optimization variants satisfy
perfect representation. We also analyze the computational complexity of
Phragm\'{e}n's methods and provide mixed-integer programming based algorithms
for computing them.
