On British railways the duty of the companies to provide all practicable safeguards and to educate and caution the servants may be said to have been faithfully performed, and the accident totals must be taken as being somewhat near the " irreducible minimum" - unless some of the infirmities of the human mind can be cured.
The first difficulty to be overcome was the algebraical solution of cubic equations, the " irreducible case" (see Equation).
Clearer ideas of imaginary quantities and the " irreducible case " were subsequently published by Bombelli, in a work of which the dedication is dated 1572, though the book was not published until 1579.
It was in seeking to close up the fissure in his system represented by this dualism that his successors succeeded only in adding weakness to weakness by reducing the principle of sufficient reason to that of formal identity (see Wolff) and representing all thought as in essence analytic. From this it immediately followed that, so far as the connexion of our experiences of the external world does not show itself irreducible to that of formal identity, it must remain unintelligible.
The individual substances, of which the universe is composed, fall into three great irreducible kinds: nature, God, man.
There remains too the consequence that there are different substances, separate from but related to one another; and these substances of three irreducible kinds, natural, supernatural, human.
The peace of Bromsebro was the first of the long series of treaties, extending down to our own days, which mark the progressive shrinkage of Danish territory into an irreducible minimum.
An irreducible element of indetermination, e.g.
Or again, within the latter in his admission of a duality of thought and " the given " in knowledge, which within knowledge was apparently irreducible, concurrently with hints as to the possibility, upon a wider view, of the sublation of their disparateness at least hypothetically and speculatively.
Indeed, if thought admits irreducible units, what can unite?
Thus in the true idea of things there is no irreducible residuum of matter: mind is the Alpha and Omega, at once the initial postulate and the final truth of reality.
It is true matter can never be completely resolved into motion, but the irreducible remainder may be treated like the 7rpc.)T fl van of Aristotle as an abstraction which we asymptotically approach but never reach.