On this mere feeling of virtual equivalence, is founded all the demonstration of those sciences, which claim the glory of being peculiarly demonstrative; our equations and proportions of abstract number and quantity involving continually this analytic valuation of notions, as reciprocally proportional. Our conception of an angle of forty-five degrees is _one_ state or affection of mind,--one state of one simple indivisible substance;--such, too, is our conception of a right angle. Our notion of _four_ or _eight_ is as much _one_ affection of mind, as our notion of a simple unit. But, in reflecting on the separate states of mind which constitute these notions, we are impressed with certain relations which they seem, to us, reciprocally to bear, and we consider the angle of forty-five degrees as equal to half the angle of ninety degrees, and our notion of eight as involving or equal to two of four. If one state of mind, which constitutes the notion of a certain abstract number or quantity, had not been considered in this sort of virtual comprehensiveness, as bearing the relation of equality, or proportion, to other states of mind, which constitute other abstract notions of the same species, mathematics would not merely have lost their certainty, but there could not, in truth, have been any such science as mathematics. The _intellectual analysis_, which appears to me to constitute so important a part of the science of mind, is nothing more than the successive developement, in application to the various mental phenomena, of this feeling of equivalence, or comprehensiveness, which is not confined to the mathematical notions of number and quantity, (though, from the greater simplicity of these, _their_ equality or proportion may be more accurately distinguished,) but extends to every thought and feeling which we regard as complex, that is to say, to almost every thought and feeling of which the mind is susceptible. We compare virtue with virtue, talent with talent, not, indeed, with the same precision, but certainly in the same manner, and with the same feeling of proportion, as we compare intellectually one angle with another; and we ask what ideas are involved in our complex notions of religion and government, with as strong a feeling that a number of ideas are virtually involved or comprehended in them, as when we ask, how often the square of two is repeated in the cube of six.