=227.= It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a court of justice, _without passion, without apology_; knowing that their reasons, as Seneca testifies of them, are not brought to _persuade_, but to compel.--BARROW, ISAAC. _Mathematical Lectures (London, 1734), p. 64._ =228.= What is exact about mathematics but exactness? And is not this a consequence of the inner sense of truth?--GOETHE. _Sprüche in Prosa, Natur, 6, 948._ =229.= ... the three positive characteristics that distinguish mathematical knowledge from other knowledge ... may be briefly expressed as follows: first, mathematical knowledge bears more distinctly the imprint of truth on all its results than any other kind of knowledge; secondly, it is always a sure preliminary step to the attainment of other correct knowledge; thirdly, it has no need of other knowledge.--SCHUBERT, H. _Mathematical Essays and Recreations (Chicago, 1898), p. 35._