Every discovery encloses a germ of hope. However great this may have been in the case of Kirchhoff, it could not by any stretch of imagination approach the degree of its fulfilment. The fundamental theoretical idea that "a vapour absorbs from the ray-complex of white light only those wave-lengths which it can emit" gave rise to a process, the ingenuity, delicacy, and certainty of which is almost inconceivable. When rays of light emitted by incandescent vapour were separated by a prism, there were discovered fine coloured lines that betrayed some unknown mystery. The spectroscopic experiments proved, in a succession of results, that the author of the above idea had made not only one discovery, but a whole host of them. For example, it was observed that, in burning minute residues obtained by evaporating certain mineral waters, a red line and a blue line that had never been seen before appeared in the spectrum. One knew immediately that an element, hitherto undiscovered, was proclaiming its presence. In this way in quick succession the element Cæsium was discovered, then Rubidium, Thallium, Indium, Argon, Helium, Neon, Krypton, Xenon—certainly things that were already pre-formed in Nature, just as the idea of a bridge from Optics to Chemistry lay all ready in the heart of Nature; but no blame can be given to the astonished contemporaries who regarded this fundamental discovery of spectroscopic analysis as a creative achievement of the intellect.
This ray of hope gave a glimpse of the degree of accuracy attainable. In this connexion the experiment confirmed infinitely more than the boldest imagination could ever have dreamed. A yellow line was detected in the spectrum of sodium. And it was found experimentally that the three-millionth part of a thousandth of a gramme of a sodium salt is sufficient to produce this sodium line in the spectrum of a Bunsen burner. There commenced a dizzying passage in the Calculus of Probabilities for, since it was found that in the sun's atmosphere hydrogen, carbon, iron, aluminium, calcium, sodium, nickel, chromium, zinc, and copper were present, the question arose as to how great was the possibility of an error in this observation. Kirchhoff calculated it as a chance of a trillion to one that these substances are actually present in the sun!
Never before had an experiment verified to such an extreme degree a discoverer's idea. It seems appropriate at this stage to deal with a doctrine which seeks to shed light into the deepest recesses of the connexion between experiment and discovery. It teaches that an experimentum crucis, an experiment that verifies absolutely, is impossible in physics. That is to say, every idea of a discoverer involves a hypothesis, and, however the experiment that follows may turn out, there still remains the possibility that this hypothesis was false, and may later have to make way for another essentially contradictory hypothesis which will be valid again only for a limited time.
The chief exponent of this theory is the eminent scholar, Pierre Duhem, Membre de l'Institut. He draws a parallel between experiment and mathematical proof, particularly with the indirect, apagogic form which has been so successfully applied in Euclidean geometry. In this method it is assumed that a certain statement is erroneous; it is then shown that it leads to an obvious contradiction; consequently the statement was correct provided that a certain doubt be excluded. Thus in the domain of mathematics we have a real experimentum crucis.
In accordance with this, Duhem tests the validity of two physical theories, both of which were put forward and claimed as discoveries. Newton had discovered the nature of light to consist in "emission"; to him, as well as to Laplace and Biot, light consists of projectiles that are emitted with very great velocity. The discovery of Huyghens, supported by Young and Fresnel, substitutes wave-motion in place of corpuscular emission. Hence, according to Duhem, we have, or we had, here two hypotheses which appear to be the only ones possible. Experiment was to pronounce a judgment, and at first it decided irrefutably in favour of the wave-theory. Therefore, the discovery of Huyghens is alone true, and that of Newton is shown to be an error; there is no third outlet, and so we have quite certainly an experimentum crucis before us.
The term itself originates in Bacon's Novum Organum. Contrary to Duhem's assumption, it does not refer to a signpost at cross-roads giving various routes, nor is it connected with croix ou pile, heads or tails. Experimentum crucis denotes rather a divine judgment at the cross, that is a test that is absolutely decisive and beyond further appeal. But no! adds Duhem, there is no room for a third judgment in the case of two contradictory statements in geometry, but there is between two contradictory statements in physics. And, in fact, this third possibility has manifested itself in the discovery of Maxwell, who has shown that the nature of light is founded on a process of periodic electromagnetic disturbances. Hence, so concludes Duhem, experiment can never decide whether a certain theory is alone valid. The physicist is never certain that he has exhausted all conceivable possibilities, of thought. The truth of a physical statement, the validity of a discovery, cannot be confirmed by any experimentum crucis.
According to this argument, therefore, it is also possible that the scientific grounds of spectral analysis do not conform to truth. A contradictory hypothesis may, indeed, be set up, with the result that the same experiments that had led Kirchhoff's discovery from one triumph to another would have to be interpreted in a totally different sense.
I must frankly confess that I cannot subscribe to such an extreme eventuality, since, in my opinion, Duhem's analogy with mathematics excludes this possibility. For if a certain probability is expressed by a trillion to one, then I venture to state that even in the case of mathematical truths certainty reaches no higher degree of probability. From the history of mathematics we know of theorems which were enunciated and provided with complete proofs, and yet did not succeed in establishing themselves; hence we see that, however evident a mathematical theorem may be, it is still only a matter of very great probability.
If, following our usual habits of thought, we take this for absolute certainty, then we may also consider the sum-total of experiments in the realm of spectral analysis to be a great experimentum crucis for the correctness of the theory itself.
Far removed from it, and yet connected with it, there is the "Periodic System of the Elements," the discovery of Mendelejew and Lothar Meyer. It, too, offered prophetic glances into the future, foretold the unknown, hinted at things that were present only in imagination in a scheme of thought that assigned definite places of existence to undiscovered things. The Periodic System is represented by a table containing vertical and horizontal rows, in the squares of which the elements are entered according to certain rules depending on their atomic weights. The discovery consisted theoretically in stating that the physical and chemical properties of each element is the arithmetic mean between the properties of its horizontal and vertical neighbours. This gave rise to predictions concerning the unoccupied squares. These gaps, these blank spaces in the table, seem to say prophetically: There are elements missing here that must be discoverable. The neighbours will betray them, and the empty space itself shows by what means they are to be found. With the shrewdness of a detective, Mendelejew was able to say: There must be elements of the atomic weights 44, 70, and 72; we do not know them yet, but we are in a position to determine the properties of these foundlings of the future, and, what is more, the properties of their compounds with other elements. Later researches, which led to the discovery of the elements. Scandium, Gallium, and Germanium, have actually confirmed all these predicted properties.