=1412.= Mathematics, considered as a science, owes its origin to the idealistic needs of the Greek philosophers, and not as fable has it, to the practical demands of Egyptian economics.... Adam was no zoölogist when he gave names to the beasts of the field, nor were the Egyptian surveyors mathematicians.--HANKEL, H. _Die Entwickelung der Mathematik in den letzten Jahrhunderten (Tübingen, 1884), p. 7._ =1413.= There are only two ways open to man for attaining a certain knowledge of truth: clear intuition and necessary deduction.--DESCARTES. _Rules for the Direction of the Mind; Torrey’s The Philosophy of Descartes (New York, 1892), p. 104._ =1414.= Mathematicians have, in many cases, proved some things to be possible and others to be impossible, which, without demonstration, would not have been believed.... Mathematics afford many instances of impossibilities in the nature of things, which no man would have believed, if they had not been strictly demonstrated. Perhaps, if we were able to reason demonstratively in other subjects, to as great extent as in mathematics, we might find many things to be impossible, which we conclude, without hesitation, to be possible.--REID, THOMAS. _Essay on the Intellectual Powers of Man, Essay 4, chap. 3._ =1415.= If philosophers understood mathematics, they would know that indefinite speech, which permits each one to think what he pleases and produces a constantly increasing difference of opinion, is utterly unable, in spite of all fine words and even in spite of the magnitude of the objects which are under contemplation, to maintain a balance against a science which instructs and advances through every word which it utters and which at the same time wins for itself endless astonishment, not through its survey of immense spaces, but through the exhibition of the most prodigious human ingenuity which surpasses all power of description.--HERBART, J. F. _Werke Kehrbach (Langensalza, 1890), Bd. 5, p. 105._ =1416.= German intellect is an excellent thing, but when a German product is presented it must be analysed. Most probably it is a combination of intellect (I) and tobacco-smoke (T). Certainly I₃T₁, and I₂T₁, occur; but I₁T₃ is more common, and I₂T₁₅ and I₁T₂₀ occur. In many cases metaphysics (M) occurs and I hold that I_{a}T_{b}M_{c} never occurs without b + c > 2a. N. B.--Be careful, in analysing the compounds of the three, not to confound T and M, which are strongly suspected to be isomorphic. Thus, I₁T₃M₃ may easily be confounded with I₁T₆. As far as I dare say anything, those who have placed _Hegel, Fichte_, etc., in the rank of the extenders of _Kant_ have imagined T and M to be identical.--DE MORGAN, A.