Scientific computing as a separate discipline is a rather recent phenomenon. It grew beyond its infancy when simulations were made possible by electronic computers. They were able to evaluated numerically rather complicated models for all kinds of applications. For many of these applications, the model is formulated as a differential equation.
Of course, like everything in mathematics, there are much older roots going back to the Babylonians. So the book starts with some ideas that already existed in antiquity with methods to construct approximations for π or the square root of 2. Somewhat more advanced numerical computations by hand were tedious but necessary in astronomical computations (Copernicus, Kepler, Galileo) and once calculus was introduced by Newton and Leibniz, derivatives and integrals, polynomials interpolation and approximation, iterative methods, and the solution of linear systems and least squares problems, became common computational issues. All this, although introduced before the computer era, they evolved little, and are still today topics in any introductory numerical analysis course with names like Newton-Raphson, Lagrange, Hermite, Gauss, Cholesky associated with these techniques.
Somewhat more advanced are Fourier series, and Fourier transforms, and elementary methods for solving differential equations (Euler, Adams-Bashford, Runge-Kutta), but still they would easily fit in a basic course. In this sense, the book evolves like a numerical course where the historical aspects are somewhat expanded at the expense of the numerical issues, but the main formulas and ideas are there. The numerics are however too thin to be useful as lecture notes for a course on numerical analysis. Therefore the book can only serve as additional literature to accompany a true course about the numerics.
The solution of differential equations, ordinary (ODE) and partial (PDE), are the main focus of the book. Still in the chapter on the pre-computer era we are exposed to PDE methods like Ritz-Galerkin, finite elements, and finite difference methods. For larger and/or nonlinear problems like weather prediction these elementary methods failed. Some short discussion on nonlinear optimization and linear programming conclude the pre-computer chapter.
With the first digital computers emerging, the numerics became more important. More complex methods were used and that required a more thorough analysis. Conditions for stability and convergence of methods for ODE and PDE were obtained and needed to be incorporated in the algorithms. Stiff ODE problems required special attention, fluid dynamics problems could have shock waves that behave like singularities and as the problems became larger, also the linear algebra involved needed stable methods (i.e. orthogonal transforms and orthogonal operations like Givens or Householder) and as the problems became really big (although often structured and/or sparse), iterative methods are more useful for linear systems than direct methods.
A major contribution of the book however, is the part discussing scientific computing as a new discipline with electronic computers fully deployed. Again methods for ODE and PDE are the main focus with further development of convergence and stability conditions, the use of new types of functions (splines, wavelets, radial basis functions), finite element methods (FEM) and spectral methods, the use of Fast Fourier Transforms (FFT), multiscale and fast multipole methods, and domain decomposition. In linear algebra, the singular value decomposition (SVD) and iterative subspace methods such as Krylov and Lanczos become widely used techniques. Some problems are really beyond reach of any deterministic method and then stochastic approximations such as Monte Carlo methods can be applied. A short concluding chapter gives a brief summary and mentions applications in physics, chemistry and engineering.
It is clear that much attention goes to the evolution in the more recent era since the digital computer is used as an essential tool. This means that the technical problems discussed are the more recent ones that usually go beyond the level of a basic introduction to numerical analysis. It is about why and how these new techniques were developed, and particularly also who was the one that did it. Several of them, mostly mathematicians and engineers, are still active in the field. So after a sweep over all the previous centuries, the emphasis is on contemporary history. Throughout the book, the numerical methods and problems are explained, but only briefly. Thus again, and even more so for the second half of the book, the reader should be familiar with the methods, and use this text as additional or background information for a course where the numerics are more thoroughly discussed. The emphasis on differential equations is a choice of the author, but that should not be surprising because differential equations are certainly a major tool in modelling a lot of problems from applied mathematics. There are broad classical application domains (structural mechanics, fluid dynamics, etc.) associated with particular types of differential equations. The methods are discussed essentially for their numerical issues, i.e., not relying on the concrete application domain. Problems of stochastic-, algebraic-, or delay-differential equations and other hybrid forms are currently hot research topics but perhaps too recent, so these are not included here.
To conclude: this book is certainly not a book to learn methods of scientific computing. The historical aspect what the title promises, and this is what you get. Those are obviously interesting, but perhaps not essential, for anyone studying the methods in a different course. It is not exactly a classical book on the history of science, partly so because it deals with a very recent evolution. Besides being of interest for current students, some of its value certainly lies also in the fact that it records this recent evolution, almost while it is happening, and saves this information about persons, methods and their sequentiality for future generations.