II. If we now consider the progress of abstraction from a more general point of view (instead of following it step by step, from its lowest to its highest degree, as in the preceding chapters), i. e., according to its orientation towards a given end, we find that it has followed three principal directions during its history: practical, speculative, scientific. These are, indeed, inseparable, inasmuch as practical abstraction leads to science, scientific abstraction is profitable to practice, and speculation cannot entirely forego the other two.[137] Abstraction and practical generalisation are necessarily the first in order, as we found in studying their first appearance in the lower animals, in children, and in savages. They serve to distinguish the qualities of things by some word, or sign; they subserve the simple adaptations of daily life. Later on, at a higher stage, they note the appearance of mixed processes, which, while more especially directed to utility, are already the prelude to scientific knowledge. Disinterested curiosity has awakened, and timidly makes for daylight. A minimum acquaintance with the history of the sciences teaches us that all were at their origin processes of _applied_ research, and that often, in their uncertain efforts, our forbears found what they were not looking for. The numerative systems issued from the need of counting objects, and later on, from rude commercial exchanges. Elementary geometry was required, in order to measure the fields, to determine a right angle, to fix relative positions, and to furnish the indispensable parts of primitive architecture. The invention of the lever, of the balance, of rudimentary engines for the lifting of heavy masses, gave the first foundations of mechanics. Astronomy arose in the desire to regulate civil life and the religious festivals, and the wish (e. g., among the Peruvians and Mexicans) not to irritate the gods by delaying the sacrifices due to them. Metallurgy, and later on the search for the philosopher’s stone, and the elixir of life, were the prelude to scientific chemistry. The historical outset of each science would furnish a profusion of similar facts. The two other operations issued by an internal division of labor from this—at first the only—tendency of the mind.