Malthus laid down the principle that populations increase in geometrical ratio, but that subsistence increases only in arithmetical ratio. He argued that a stage is reached where increase of populations must be limited by sheer want, and he advocated checks on population increase in order to reduce misery and want. His work was an important influence on both Darwin and Wallace in their formulation of the concept of natural selection. It also had a profound influence on the decrease in size of families down to the present time. The book was at first published anonymously, but Malthus attached his name to the greatly expanded second edition of 1803. Malthus continued to revise the work through the sixth edition, 2 vols., 1826. All editions but the fourth contain significant new material.