The introduction and the last section illustrate clearly how this paper relates to the literature. The authors give a general picture of the stream of research that evolved from the publication of the Karmarkar method in 1984 and resulted in the development of a number of so-called interior point methods. Such methods are discovered to be strictly interrelated. For example, in a previous paper of the second author, it is stated that ``the projective algorithms for a phase II problems are all the same''. The method proposed here is essentially that of \textit{C. C. Gonzaga} [SIAM Rev. 34, No. 2, 167-224 (1992; Zbl 0763.90063)] and hence belongs to the category of path following methods. So, where is the contribution? It is in the convergence analysis. Providing a simple and elegant proof of the polynomial complexity of the method is the main topic of the paper. The method has its limitations --- the feasible regions of both the problem and its dual must have non-empty interiors and the coefficient matrix full row rank (although this latter hypothesis is claimed to be not essential). It is interesting to note that, in contrast with the promise of developing, in a subsequent note, some trickery that will lead to a record complexity, the method is deluding, when it comes to practical efficiency. The remedies are in the spirit of other already proposed methods. For the high quality of this paper, it leaves the desire of refreshing novelty in either the approach or the class of problems dealt with. This stream of research seems to be close to saturate its potential. Fortunately a revival of interest toward nonlinear optimization is emerging within the field of interior point methods.