We define two notions of partial sums of a Riordan array, corresponding respectively to the partial sums of the rows and the partial sums of the columns of the Riordan array in question. We characterize the matrices that arise from these operations. On the one hand, we obtain a new Riordan array, while on the other hand, we obtain a rectangular array which has an inverse that is a lower Hessenberg matrix. We examine the structure of these Hessenberg matrices. We end with a generalization linked to the Fibonacci numbers and phyllotaxis.