I cannot assent, indeed, to the common notion that certainty begins and ends with numerical determination. Nothing is more certain than logical truth. The laws of identity and difference are the tests of all that is certain throughout the range of thought, and mathematical reasoning is cogent only when it conforms to these conditions, of which logic is the first development. And if it be erroneous to suppose that all certainty is mathematical, it is equally an error to imagine that all which is mathematical is certain. Many processes of mathematical reasoning are of most doubtful validity. There are points of mathematical doctrine which must long remain matter of opinion; for instance, the best form of the definition and axiom concerning parallel lines, or the true nature of a limit. In the use of symbolic reasoning questions occur on which the best mathematicians may differ, as Bernoulli and Leibnitz differed irreconcileably concerning the existence of the logarithms of negative quantities.[87] In fact we no sooner leave the simple logical conditions of number, than we find ourselves involved in a mazy and mysterious science of symbols. [87] Montucla. *Histoire des Mathématiques*, vol. iii. p. 373. Mathematical science enjoys no monopoly, and not even a supremacy, in certainty of results. It is the boundless extent and variety of quantitative questions that delights the mathematical student. When simple logic can give but a bare answer Yes or No, the algebraist raises a score of subtle questions, and brings out a crowd of curious results. The flower and the fruit, all that is attractive and delightful, fall to the share of the mathematician, who too often despises the plain but necessary stem from which all has arisen. In no region of thought can a reasoner cast himself free from the prior conditions of logical correctness. The mathematician is only strong and true as long as he is logical, and if number rules the world, it is logic which rules number. Nearly all writers have hitherto been strangely content to look upon numerical reasoning as something apart from logical inference. A long divorce has existed between quality and quantity, and it has not been uncommon to treat them as contrasted in nature and restricted to independent branches of thought. For my own part, I believe that all the sciences meet somewhere. No part of knowledge can stand wholly disconnected from other parts of the universe of thought; it is incredible, above all, that the two great branches of abstract science, interlacing and co-operating in every discourse, should rest upon totally distinct foundations. I assume that a connection exists, and care only to inquire, What is its nature? Does the science of quantity rest upon that of quality; or, *vice versâ*, does the science of quality rest upon that of quantity? There might conceivably be a third view, that they both rest upon some still deeper set of principles.