=1308.= Confined to its true domain, mathematical reasoning is admirably adapted to perform the universal office of sound logic: to induce in order to deduce, in order to construct.... It contents itself to furnish, in the most favorable domain, a model of clearness, of precision, and consistency, the close contemplation of which is alone able to prepare the mind to render other conceptions also as perfect as their nature permits. Its general reaction, more negative than positive, must consist, above all, in inspiring us everywhere with an invincible aversion for vagueness, inconsistency, and obscurity, which may always be really avoided in any reasoning whatsoever, if we make sufficient effort.--COMTE, A. _Subjective Synthesis._ =1309.= Formal thought, consciously recognized as such, is the means of all exact knowledge; and a correct understanding of the main formal sciences, Logic and Mathematics, is the proper and only safe foundation for a scientific education.--LEFEVRE, ARTHUR. _Number and its Algebra (Boston, Sect. 222.)_ =1310.= It has come to pass, I know not how, that Mathematics and Logic, which ought to be but the handmaids of Physic, nevertheless presume on the strength of the certainty which they possess to exercise dominion over it.--BACON, FRANCIS. _De Augmentis, Bk. 3._ =1311.= We may regard geometry as a practical logic, for the truths which it considers, being the most simple and most sensible of all, are, for this reason, the most susceptible to easy and ready application of the rules of reasoning.--D’ALEMBERT. _Quoted in A. Rebière: Mathématiques et Mathématiciens (Paris, 1898), pp. 151-152._ =1312.= There are notable examples enough of demonstration outside of mathematics, and it may be said that Aristotle has already given some in his “Prior Analytics.” In fact logic is as susceptible of demonstration as geometry, .... Archimedes is the first, whose works we have, who has practised the art of demonstration upon an occasion where he is treating of physics, as he has done in his book on Equilibrium. Furthermore, jurists may be said to have many good demonstrations; especially the ancient Roman jurists, whose fragments have been preserved to us in the Pandects.--LEIBNITZ, G. W. _New Essay on Human Understanding [Langley], Bk. 4, chap. 2, sect. 12._