=1945.= The _infinite_ divisibility of _finite_ extension, though it is not expressly laid down either as an axiom or theorem in the elements of that science, yet it is throughout the same everywhere supposed and thought to have so inseparable and essential a connection with the principles and demonstrations in Geometry, that mathematicians never admit it into doubt, or make the least question of it. And, as this notion is the source whence do spring all those amusing geometrical paradoxes which have such a direct repugnancy to the plain common sense of mankind, and are admitted with so much reluctance into a mind not yet debauched by learning; so it is the principal occasion of all that nice and extreme subtility which renders the study of Mathematics so difficult and tedious.--BERKELEY, G. _On the Principles of Human Knowledge, Sect. 123._ =1946.= To avoid misconception, it should be borne in mind that infinitesimals are not regarded as being actual quantities in the ordinary acceptation of the words, or as capable of exact representation. They are introduced for the purpose of abridgment and simplification of our reasonings, and are an ultimate phase of magnitude when it is conceived by the mind as capable of diminution below any assigned quantity, however small.... Moreover such quantities are neglected, not, as Leibnitz stated, because they are infinitely small in comparison with those that are retained, which would produce an infinitely small error, but because they must be neglected to obtain a rigorous result; since such result must be definite and determinate, and consequently independent of these _variable indefinitely small quantities_. It may be added that the precise principles of the infinitesimal calculus, like those of any other science, cannot be thoroughly apprehended except by those who have already studied the science, and made some progress in the application of its principles. --WILLIAMSON, B. _Encyclopedia Britannica, 9th Edition; Article “Infinitesimal Calculus,” Sect. 12, 14._ =1947.= We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads. --VOLTAIRE. _A Philosophical Dictionary; Article “Infinity.” (Boston, 1881)._