Book excerpt: The investigation was concerned with the replacement of physical entities by models, upon which calculations may be performed; often the aim is to interpret physical conditions of the physical entity as mathematical conditions on some part of a mathematical model, or vice versa. The physical entities considered were time-varying electrical networks, which were often assumed to be imbued with physical properties such as linearity, passivity, and finiteness, all properties well known in more classical studies; the mathematical models considered were principally described by network matrices, such as the scattering matrix or impedance matrix. One aim was necessary, sufficient, or necessary and sufficient conditions on a scattering matrix for the associated network to be linear, or passive, or finite, or to possess any combination of these and other properties. The principal aim was means for passing from the network to a matrix description (analysis), and means for passing from a matrix to the network (synthesis).