excused; but all errors are productive, and it should be remembered that this one has produced that extraordinary perversion of truth to be found in the reply to the question, How is all this confusion to be brought into harmony? It has produced it not only in words, but in deed. Was it not this reply that led the French commission to extend the use of the present base from numeration to gradation also, under the delusive hope of producing a perfectly homogeneous system, that would be practical also? Was it not under its influence, that, adhering to the base to which the world had been so long accustomed, instead of attempting to regulate ideal division by real, which might have led to the adoption of the true base and a practical system, they committed the one great error of endeavoring to reverse true order, by forcing real division into conformity with a preconceived ideal? This attempt was made at a time supposed by many to be peculiarly suited to the purpose, a time of changes. It was a time of changes, truly; but these were the result of high excitement, not of quiet thought, such as the subject requires,--a time for rushing forward, not for retracing misguided steps. Accordingly, a system was produced which from its magnitude and importance was truly imposing, and which, to the present day, is highly applauded by all those who, under the influence of the error alluded to, conceive decimal numeration to be a sacred truth: applauded, not because of its adaptation to commerce, but simply because of its beautiful proportions, its elegant symmetry, to say nothing of the array of learning and power engaged in its production and inauguration: imposing, truly, and alike on its authors and admirers; for the qualities they so much admire are not peculiar to the decimal base, but to the use of one and the same base for numeration, notation, and gradation. But if the base ten agrees with nothing, over, on, or under the earth, can it be the best for scientific use? can it be at all suited to commercial purposes? If true order is the object to be attained, and that for the sake of its utility, then agreement between real and ideal division is the one thing needful, the one essential change without which all other changes are vain, the only change that will yield the greatest good to the greatest number,--a change, which, as volition is with the ideal, and inertia with the real, can be attained only by adaptation of the ideal to the real. A full investigation of the existing heterogeneous or fragmentary system will lead to the discovery that it contains two elements which are at variance with natural division and with each other, and that the unsuccessful issue of every attempt at regulation hitherto made has been the proper result of the mistake of supposing agreement between those elements to be a possible thing.