_Lectures on Shakespere (Bohn Library), p. 52._ =535.= I have come to the conclusion that the exertion, without which a knowledge of mathematics cannot be acquired, is not materially increased by logical rigor in the method of instruction. --PRINGSHEIM, ALFRED. _Jahresbericht der Deutschen Mathematiker Vereinigung (1898), p. 143._ =536.= The only way in which to treat the elements of an exact and rigorous science is to apply to them all the rigor and exactness possible.--D’ALEMBERT. _Quoted by De Morgan: Trigonometry and Double Algebra (London, 1849), Title page._ =537.= It is an error to believe that rigor in proof is an enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.--HILBERT, D. _Mathematical Problems; Bulletin American Mathematical Society, Vol. 8, p. 441._ =538.= Few will deny that even in the first scientific instruction in mathematics the most rigorous method is to be given preference over all others. Especially will every teacher prefer a consistent proof to one which is based on fallacies or proceeds in a vicious circle, indeed it will be morally impossible for the teacher to present a proof of the latter kind consciously and thus in a sense deceive his pupils. Notwithstanding these objectionable so-called proofs, so far as the foundation and the development of the system is concerned, predominate in our textbooks to the present time. Perhaps it will be answered, that rigorous proof is found too difficult for the pupil’s power of comprehension. Should this be anywhere the case,--which would only indicate some defect in the plan or treatment of the whole,--the only remedy would be to merely state the theorem in a historic way, and forego a proof with the frank confession that no proof has been found which could be comprehended by the pupil; a remedy which is ever doubtful and should only be applied in the case of extreme necessity. But this remedy is to be preferred to a proof which is no proof, and is therefore either wholly unintelligible to the pupil, or deceives him with an appearance of knowledge which opens the door to all superficiality and lack of scientific method.--GRASSMANN, HERMANN. _Stücke aus dem Lehrbuche der Arithmetik; Werke, Bd. 2 (Leipsig, 1904), p. 296._