Gaussian objectives were now frequently produced and calculated with other kinds of glass—apparently with varying success. According to Steinheil of Munich it was done for the first time in England, but the result was bad. In 1860 Steinheil himself successfully produced a Gauss objective. Later experts were not of one opinion as to the relative value of the Gauss and Fraunhofer objectives. The Gauss objective met its greatest success in microscopes. At the close of his paper on achromatic double objectives Gauss discussed the correction of spherical aberration.
About 1840 Gauss returned to the question of constructing achromatic objectives, when the Viennese mechanic and optician Simon Plössl began constructing his so-called dialytic telescopes. This type owes its origin to the difficulty of producing large clear pieces of flint glass. The French and English governments had offered large prizes for the solution of the problem.
Gauss stated in a letter to Encke, dated January 2, 1840, that the two conditions for complete achromatism are (a) that the red and the blue image fall on a plane normal to the telescope axis, and (b) that they be of the same size. It was his recognition of b that made him so skeptical of the dialytic telescope. He felt that in three lenses far apart the second condition of achromatism could be fulfilled, but not if two of them, as was the case in the dialytic telescopes, are cemented in and are close together.
In the fall of 1840 both Encke and Schumacher sent Gauss a dialytic telescope for study. Encke was so affected by Gauss’ skepticism that he was ready to drop his original view about the excellence of Plössl’s instruments. We can imagine Encke’s surprise when he received Gauss’ letter of December 23, 1840, which informed him that the chromatic error of the dialytic objective can be completely compensated by the eyepiece, so that the eye receives an image completely free of color. This principle is used today in the so-called compensation eyepieces of modern microscopes. Gauss was thus finally convinced of the advantages of the Plössl instruments and purchased one in the spring of 1841; he was pleased with its performance.
Scattered through Gauss’ correspondence from 1813 to 1846 is the discussion of many matters in optics which are considered very elementary today. However, it is well to remember that such topics were not so well understood by physicists of his day. They deal with general properties of the course of rays in telescopes, such as magnification, brightness, modification for nearsighted eyes, and so forth.44
The so-called Gauss eyepiece is still used today for the purpose of autocollimation. In spectrometers and refractometers it sets the axis of a telescope accurately at right angles to a plane polished surface. The Gauss eyepiece tube has an aperture in the side through which light is admitted to a piece of plane unsilvered glass at an angle of 45 degrees to the axis of the telescope. The light is thus reflected past the cross wires and down the telescope tube to the plane polished surface. If the latter is exactly at right angles to the telescope axis, the light will be reflected back down the telescope, and an image of the cross wires will be formed exactly coincident with the cross wires themselves. The observer must adjust the position of the telescope until this coincidence is obtained. On October 31, 1846, Gauss wrote to Schumacher: “The point is, that no glass must be between the mirror inclined at 45° and the cross-wire system.”
Of special importance was a prize problem of the Royal Society of Sciences in Göttingen for November, 1829, proposed by Gauss in the mathematical class of the society, and touching a method for photometry of the stars. The society did not grant the prize to any of the papers sent in, and renewed the problem at the insistence of Gauss, who placed at the disposal of Gerling his own ideas on such a photometer, so that the latter could work out his thoughts on the subject and compete for the prize. As a matter of fact. Gauss was interested in the problem of heterochromatic photometry. Actually, the problem was not solved until 1920 by Schrödinger. Gerling constructed a photometer according to the principle indicated by Gauss, but he merely got honorable mention. The prize went to Steinheil of Munich who participated in the competition with a photometer based on other principles of construction.
By far Gauss’ greatest achievement in the field of optics was his Dioptrische Untersuchungen, which appeared in 1840. According to his own claim he had possessed the results for forty or forty-five years, but had always hesitated to publish such elementary meditations. A work of Bessel on the determination of the focal distance of the Königsberg heliometer objective gave him the impetus necessary to publication. Bessel’s method assumed mistakenly that the usual lens formula
is correct for lenses of finite thickness. As a consequence of this mistake, Bessel greatly underestimated the possible error of his measurement. Dioptrische Untersuchungen treats the problem of pursuing the course of a ray of light through a centered system of refracting spherical surfaces. The equations of the ray before the first refraction are to be set in relation to the equations of the ray after the last refraction, that is, the coefficients of the latter equations are to be deduced from those of the former.
In the Dioptrische Untersuchungen there are data on the construction of the image when the principal points and foci of the system are given, and finally formulas for a simple lens of nonvanishing thickness are given. Bessel’s determination of the focal distance of the Königsberg heliometer objective was examined. While Bessel estimated the error of his result at 1/75,000, Gauss showed that it amounted to 1/1,300.
Gauss wrote in unpublished notes that reflections on spherical surfaces are to be incorporated into his theory by making the index of refraction negative. The light rays fall on a lens, are refracted the first time on the front surface, reflected at the rear surface, and refracted again at the front surface. His formulas indicate principal points and foci for this case.
The Dioptrische Untersuchungen emphasized that the position of the principal points and foci depends on the index of refraction of the lenses of the system, that is, it varies from wave length to wave length, and that in general chromatic aberration occurs. For achromatism he demanded that all parallel rays independently of color converge at one point, that is, not only such as are parallel to the axis, but such as are inclined to it. In the usual achromatic objectives, in which lenses are close to each other, these conditions are approximately fulfilled, but not in the case of dialytic objectives. This explains why Gauss had doubts about the dialytic principle.
Gaussian dioptrics represents the perfection of those investigations which relate to central rays (paraxial rays), that is, to the point by point projection of means of narrow pencils of rays. In the century which has passed, practically nothing has been added to the Gaussian theory. The importance of his work is slight for practical optics today, since one cannot be limited to narrow pencils of rays, but must project by means of wide open pencils of rays.
One can arrive at the fundamental concepts and properties of projection in a purely geometrical manner. Gauss’ deductions produce the impression that all these concepts are limited to narrow pencils of rays, yet a geometric study shows that the relationship and these fundamental concepts are possible in wide open pencils of rays, while with Gauss only the physical production of such relationship is limited to narrow pencils.
Gauss stated that his interest in dioptrics reached back to about 1800. Nothing has been found in his manuscripts to offer proof of this statement. However, some notes on a subject closely related to his optical research date from the period 1814–1817. In them he treated fully the systems of principal rays and aperture rays. A note of 1811 dealt with the experiments of Malus on the polarization of light.
In 1836 Gauss purchased a Schwerd instrument, in order to make diffraction experiments. For a very brief time he was intensely interested in the subject, but had to give it up for lack of time and accomplished nothing in this field.
XV
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Germination: Non-Euclidean Geometry
No other phase of Gauss’ career has aroused as much controversy as his work in non-Euclidean geometry.
There are several reasons for this. Each of the three men (Lobachevsky, J. Bolyai, and Riemann) generally credited with being the founders of non-Euclidean geometry had either direct or indirect connections with Gauss. Felix Klein made some rather exaggerated statements concerning the influence of Gauss on these men, and since Gauss’ manuscripts were not open to scholars45 until very recent years, no final verdict could then be reached. Gauss himself published practically nothing on the subject, and scholars were reduced to the examination of certain passages in his letters and several book reviews where he alluded to the possibility of such systems of geometry. He was extremely sensitive to public criticism and unwilling to have anything so revolutionary appear over his name.
Critics, both contemporary with him and in our own time, have attacked Gauss’ character because he so often stated that the result of some new discovery had been in his possession for many years. They did not realize that in his early years such a flood of new developments came to his mind that he could hardly control them, and that it took him long years of work before they fully matured. He felt that a publication should be a “completed work of art,” and in fact was not too much concerned when (if ever) many of his results were published. To him, the main concern was his occupation with new truth. His similarity to Newton in this respect is striking.
Since he adhered to the above viewpoint, he was not particularly concerned when someone anticipated him by prior publication. His goal was primarily the extension and establishment of truth, not personal glory. Since his death, a careful study of his papers and letters has disclosed that his statements about early discovery were correct. In addition, one should remember that Gauss had an unusually accurate memory. He used notes of the years 1796–1815 in his scientific diary, which was not “discovered” until 1898. Later in life he entered in notebooks mathematical results which he had mentioned in letters. Thus it has been possible to compare his utterances with the evolution of his thoughts.
On November 28, 1846, Gauss wrote to Schumacher that in 1792 at the age of fifteen he had thought of a geometry “which would have to occur and would occur in a rigorously consistent manner, if Euclidean geometry is not the true one,” that is, if the eleventh axiom (of parallels) is not valid. Of course, Gauss meant merely the first dawning of the thought. On October 2, 1846, he had told Gerling that the theorem that in every geometry independent of the parallel axiom the area of a polygon is proportional to the deviation of the sum of the external angles from 360 degrees is “the first theorem (likewise at the threshold) of the theory, which I recognized in 1794 as necessary.”
We know that Gauss was giving much thought to the possibility of non-Euclidean geometry in the years 1797–1802. In his diary an entry in September, 1799, contains these words: “In principiis geometriae egregios progressus fecimus.” On May 17, 1831, Gauss reported to Schumacher that he had begun to write up some of his meditations on parallel lines, which in part were about forty years old. Germination, therefore, occurred in the years 1792–1794.
When Gauss began his studies in Göttingen in October, 1795, he had recognized the weak position of the Euclidean system, and m the case of polygon areas had considered the consequences “Which arise from a rejection of the parallel axiom. This type of meditation” was in the air at that time. There was a flood of publications on the question of parallels, especially in France in the years after the Revolution.
Legendre, who had a strong dislike for Gauss, tackled the parallel problem, but was unable to solve it. In a letter to Olbers dated July 30, 1806, Gauss remarked that it seemed to be his fate in almost all his theoretical works to be competing with Legendre, and mentioned the theory of numbers, transcendental functions, elliptic functions, the foundations of geometry, and the method of least squares. One might add to this list geodesy and the attraction of homogeneous ellipsoids.
At the University of Göttingen there was lively interest in the question of parallels. Kästner had eagerly collected literature on the subject and in 1763 had directed a worth-while dissertation of his pupil Klügel on previous attempts to prove the axiom. Kästner believed that no one in his right mind would attack Euclid’s axiom. Gauss’ friend Pfaff believed that the only thing to do was to replace the parallel axiom by a more simple one. When Gauss went to Göttingen, J. Wildt (1770–1844) gave a trial lecture on the theory of parallels (1795). In 1800 he published three “proofs” of the eleventh axiom, and in 1801 Seyffer, the professor of astronomy, published two reviews of attempts to prove the parallel axiom. Seyffer had come to the conclusion that it was more than doubtful, perhaps impossible, to prove the eleventh axiom without drawing aid from a new axiom. Gauss was not on intimate terms with Kästner and Wildt, but he was very close to Seyffer, and their correspondence continued until the latter’s death. Their conversations frequently touched on the theory of parallels.
At Seyffer’s home Gauss met Wolfgang Bolyai, the young Hungarian student who became the most intimate friend of his entire life. Needless to say, the theory of parallels was one of their chief mutual interests. After Gauss returned home to Brunswick in 1798, Bolyai made efforts to prove the axiom and in May, 1799, believed he had reached his goal. On May 24, 1799, the two said farewell forever at Clausthal in the Harz mountains; Bolyai was returning to his native Hungary. He told Gauss of his “Göttingen theory of parallels.” In a letter to Bolyai dated December 16, 1799, Gauss regretted that he did not have time to find out more about Bolyai’s work on the foundations of geometry. He felt that it would have saved him much toil, and that much remained to be done on the subject. Gauss stated in the letter that he was far advanced in this work, but that he did not have time to work it out properly. He felt that his work would make the truth of geometry doubtful. Gauss was not satisfied with what had been done; he felt that all so-called proofs of the parallel axiom were failures (1799). No notes exist of the work which Gauss had done up until that time. Certainly he recognized that the Euclidean was not the only possible system.
The works of Saccheri (1733) and Lambert (1766) on the theory of parallels were available to Gauss at the University of Göttingen library. Indeed, the record shows that he drew out volumes by Lambert in 1795 and 1797. These works were probably also known to Wolfgang Bolyai. In later years Lambert’s theory of parallels was discussed among Gauss’ pupils. Gauss owned the Mathematische Abhandlungen of J. W. H. Lehmann (Zerbst, 1829) in which Saccheri and Lambert are quoted. Marginalia and traces of use show that Gauss read it and paid special attention to the passages on parallel theory. Neither Lambert nor Saccheri was able to reach the level of non-Euclidean plane trigonometry.
Bolyai reached home in Transylvania in the summer of 1799, but had no time for mathematics until 1804, when he was appointed to a professorship of mathematics and physics at Maros-Vásárhely. He pulled out his “Göttingen theory of parallels,” polished it up, and sent the sketch to Gauss on September 16, 1804, asking for criticism and forwarding of it to some reputable scientific society for judgment.
On November 25, 1804, Gauss replied that he was delighted by the genuine, fundamental ingenuity of the little work. He stated that the train of thought was very similar to his own, but that up until that time he had been unable to solve the problem completely. Bolyai’s procedure did not satisfy him fully, but he wrote that he hoped to break through the difficulties before his death. He added that he was too busy with other matters at the time to give attention to it. Gauss promised that if Bolyai succeeded in surmounting all the hindrances, he would be glad to be anticipated by so intimate a friend and would do all in his power to make the work properly known to the public.
Bolyai was greatly encouraged by the letter, and on December 27, 1808, sent Gauss a supplement. To this he got no answer, and their correspondence was broken off until 1816. In those years Bolyai had put up a hard fight against the problem, and finally was so discouraged that he felt all his effort had been wasted.
When his son Johann Bolyai (1802–1860), who had entere
d the Academy of Military Engineers in Vienna in 1818, reported to his father in 1820 that he was attempting to prove the eleventh axiom, the elder Bolyai was horrified and in the most moving terms begged his son to leave the theory of parallels in peace:
Do not lose one hour on that. It brings no reward, and it will poison your whole life. Even through the pondering of a hundred great geometers lasting for centuries it has been utterly impossible to prove the eleventh without a new axiom. I believe that I have exhausted all imaginable ideas. Furthermore, if Gauss too had spent his time with puzzling over the eleventh axiom, his theories of polygons, his Theoria motus corporum coelestium, and all his other works would not have appeared, and he would have lagged behind. I can prove in writing, that he racked his brains on parallels. He averred both orally and in writing that he had meditated fruitlessly about it.
Johann was not terrified by his father’s warning; on the contrary, it spurred him on to action. Toward the end of 1823 he began to sense victory. His unexpected solution was that which Gauss had attained in 1816 after long hesitation and doubt.
In the letter of November 25, 1804, Gauss had indicated that his train of thought was quite similar to that of the elder Bolyai. The latter had considered the line which originates when one erects perpendiculars of the same length, on the same side, at points equally distant from a straight line, the end points being connected by a straight line. In Euclidean geometry one thus gets a parallel as a base line, but non-Euclidean geometry yields a broken line which consists of equally long segments approaching each other at equal angles. Wolfgang Bolyai had attempted to prove that a broken line of this kind, if one moves out on it far enough, would have to intersect the base line. That would prove that the assumption that the eleventh axiom is not valid leads to a contradiction. Gauss was making attempts in the same direction, as shown by the last page of his notebook “Mathematische Brouillons” (Collected Works, VIII, 163). These notes were begun in October, 1805.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 21