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.
It is natural that the range of individual experience should be very limited in comparison with the knowledge indispensably needed for acquiring an adequate conception of the world in which we live. We have, to a great extent, to rely on statements of facts which we ourselves have not observed. To enrich and to enlarge our own experience we have to imbibe the experience of others. Sometimes we can, but sometimes we cannot, verify what we have been told. For instance, that stones fall through empty space with a velocity of 32·18 English feet at the end of the first second can be verified by experiment, i. e., the experiment can be repeated under the same circumstances. But historical data such as whether Buddha died under a fig-tree, or whether Christ was crucified under Pontius Pilate, cannot be verified by experiment. Historical data are statements, not of general truths, but of single facts, which, if they are accepted at all, have to be taken on authority. The authority may be weak or strong; it may be strong enough to be equivalent practically to a certainty, which latter case occurs, for instance, when the fact in question in its direct consequences perceptibly affects our life, and its causal connection can thus be directly and indubitably traced.