developed relations between certain definite integrals (_Comptes Rendus_ of the Paris Academy, Vol. 77, 1873). Proceeding from the relations thus established, Professor Lindemann first demonstrates the following proposition: If the coefficients of an equation of _n_th degree are all real or complex whole numbers and the n roots of this equation _z_{1}, _z_{2}, ..., _z_{_n_} are different from zero and from each other it is impossible for _e_^_z_{1} + _e_^_z_{2} + _e_^_z_{3} ... + _e_^_z_{_n_} to be equal to _a_/_b_, where _a_ and _b_ are real or complex whole numbers. It is then shown that also between the functions _e_^{_rz_{1}} + _e_^{_rz_{2}} + _e_^{_rz_{3}} + ... _e_^{_rz_{_n_}}, where _r_ denotes an integer, no linear equation can exist with rational coefficients variant from zero. Finally the beautiful theorem results: If _z_ is the root of an irreducible algebraic equation the coefficients of which are real or complex whole numbers, then _e_^_z_ cannot be equal to a rational number. Now in reality _e_^{t√-1} is equal to a rational number, namely,-1. Consequently, π√-1, and therefore π itself, cannot be the root of an equation of _n_th degree having whole numbers for coefficients, and therefore also not of such an equation having rational coefficients. The property last mentioned, however, π would have if the squaring of the circle with ruler and compasses were possible. #The verdict of mathematics.# "It is impossible with ruler and compasses to construct a square equal in area to a given circle." These are the words of the final determination of a controversy which is as old as the history of the human mind. But the race of circle-squarers, unmindful of the verdict of mathematics, that most infallible of arbiters, will never die out so long as ignorance and the thirst for glory shall be united. HERMANN SCHUBERT. THE CRITERION OF TRUTH. A DISSERTATION ON THE METHOD OF VERIFICATION. Modern science rests upon the recognition of the truth that all knowledge is a statement of facts. The formulation of natural laws is nothing but a comprehensive description of certain kinds of natural processes. Natural laws are generalisations of facts. Similarly, any philosophical theory is, or from the modern standpoint ought to be, simply a systematised representation of facts. Facts are the bottom-rock to which, everywhere, we have to go down. The recognition of this maxim is called, most appropriately, positivism; and I take it that as a matter of principle all modern thinkers can and perhaps do agree to it. A Roman Catholic philosopher may consider some things as facts which a scientist of heretic England, for instance, does not; yet it is from facts, or what is thought to be facts, that every one derives his conception of the world.