1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing. 2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general. 3. The demonstration is strictly logical, taking nothing for granted except self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion. 4. When the conclusion is obtained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if, as was said before, reason is not to be the instructor, but the pupil. 5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. Between the meaning of terms there is no distinction, except a total distinction, and all adjectives and adverbs expressing difference of degrees are avoided.--DE MORGAN, AUGUSTUS. _On the Study and Difficulties of Mathematics (Chicago, 1898), chap. 1._ =431.= The instruction of children should aim gradually to combine knowing and doing [Wissen und Können]. Among all sciences mathematics seems to be the only one of a kind to satisfy this aim most completely.--KANT, IMMANUEL. _Werke [Rosenkranz und Schubert], Bd. 9 (Leipzig, 1838), p. 409._ =432.= Every discipline must be honored for reason other than its utility, otherwise it yields no enthusiasm for industry. For both reasons, I consider mathematics the chief subject for the common school. No more highly honored exercise for the mind can be found; the buoyancy [Spannkraft] which it produces is even greater than that produced by the ancient languages, while its utility is unquestioned.--HERBART, J. F. _Mathematischer Lehrplan für Realgymnasien, Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 167._ =433.= The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason.--CHANCELLOR, W. E.