What is the triangle in the purely ideal order? A creation of the understanding, which disposes the lines in a triangular form, and, preserving this form, modifies it in a thousand ways. Thus far there is only one postulate and different combinations of it: but the properties of the triangle flow by absolute necessity from the conditions of the postulate: the understanding, however, does not make these properties, it discovers them. The example of the triangle is applicable to all geometry. The understanding takes a postulate; this is its free work, but it must not come in conflict with the principle of contradiction. From this postulate flow absolutely necessary consequences, independent of intellectual action, and involving an absolute truth known by the understanding itself. Consequently it is false to say of them that it makes them. Suppose a man so to place a body, that, left to itself, it will fall to the ground: is it the man who gives it the force to fall? Certainly not, but nature. The man only supplies the condition necessary for the force of gravity to produce its effect: when once the condition is performed, the fall is inevitable. Here, then, is a simile which shows clearly and exactly what happens in the purely ideal order. The understanding performs the conditions; from them flow other truths, _not made_, but known, by the understanding. This truth is absolute, is as the force of gravity in the order of ideas. Hence we see what is admissible, and what inadmissible in Vico's system. The power of combination, a generally recognized fact, is admissible; the exaggeration of this fact extended to all truths, when it only comprises postulates in their various combinations, is inadmissible. The rules of algebra are conventional inasmuch as they relate to the _expression_, for this might evidently have been different. Supposing, however, the expression, the development of the rules, is not conventional, but necessary. In the expression a^n/a^n the number of times the quantity has entered as factor might clearly have been expressed in infinite ways; but supposing the present to have been adopted, the rule is not conventional, but absolutely necessary; since whatever the expression, it is always certain that the division of a quantity by itself, with distinct exponents, gives for result the diminution of the number of times it has entered as factor: this is denoted by the remainder of the exponents; and consequently if the number of times be equal in the dividend and the divisor, the result will be = 0. Thus we see that even in algebra, what the understanding has to do, is to perform the conditions, and express them as seems to it best: but here its free work ends, for necessary truths result from these conditions; and these it does not make, but only knows.