NON-EUCLIDEAN GEOMETRY. W. H. S. MONCK. FOR POPULAR ASTRONOMY, The question of the universal validity of Euclid's axiom regarding parallel lines was one which exercised a good deal of my attention many years ago when I was a student. I came to the conclusion that strict proof was impossible, but that the axiom itself was true and that the objections to it arose either from not realizing its real import in the imagination, or from losing sight of the fact that we were dealing with the real properties of space and that, as always happens when we apply algebraic or symbolical reasonings to real objects, we might readily carry this symbolism too far. At this time I took no interest in astronomy. Indeed the book then used in Dublin University would hardly have led any one to think that astronomy was a physical science of a very progressive character. But one thing it seemed to me that Legendre did establish without the aid of Euclid's famous axiom viz: that the three angles of a plane triangle could not exceed two right angles. The difficulty was to prove that they could not be less-perhaps even less than one right angle. The proof that they cannot be greater is substantially the following: A B D C Let ABC be any triangle and let Cbe its greatest angle and B the next in magnitude. Draw AD bisectEing BC and make DE equal to AD and join CE. Now the triangle EDC is evidently equal in all respects to the triangle ADB, and the sum of the three angles of the triangle AEC is evidently equal to that of the original triangle ABC. But the angle ACE is equal to the sum of the two greates tangles of the original triangle ABC viz: ACB and ABC. We can repeat this process as often as we like (bisecting AC instead of CE next time, if that course will increase the angle at C more rapidly), and by carrying this proces on sufficiently far we can obtain a triangle the sum of whose three angles is the same as that of the original triangle, but the two smaller angles in which taken together do not amount to one-millionth (or one hundredth millionth) part of the smallest of the three angles of ABC. It seemed to me also that the failure to prove Euclid's axiom in the terms laid down arose from the difficulty of applying the principle of superposition to infinites. Let us suppose a number of right lines drawn from a centre and produced to infinity, the angles between each successive pair being equal. Then if the principle of superposition were applied each adjacent pair of lines would enclose an equal space, and if there were a million such lines each pair would enclose one millionth part of the whole plane extended to infinity. Now on the contrary take a pair of right lines crossing a third in such a manner that the two interior angles are together equal to two right angles. = Let AB* be the intersected line and let the two intersecting lines be CF and DG. Take DE CD and draw EH making the angle DEH the angle CDG. Suppose all four lines produced to infinity towards the right. Then applying the principal of superposition the space enclosed between the CF and DG and that enclosed between the lines DG and EH (to the right of the intersecting line AB) will be equal. But if AB be produced to infinity we can mark off as many segments equal to CD upon it as we please and draw lines under similar conditions through the concluding point of each segment. Consequently the space included between the pair of lines CF, DG produced to infinity and the intersecting line AB does not bear any finite proportion to the whole plane extended to infinity. Then draw CI such that the two angles CDG, DCI are together less than two right angles, reduce all the lines to infinity. The space included in the angle FCI is greater than that included in between the lines CF and DG since the former bears a finite ratio to the entire plane extended to infinity while the latter does not. But if the line CI does not intersect the line DG the former space is included in the latter space and we should have a part greater than the whole. Consequently if the principle of superposition is applicable to infinity Euclid's axiom holds good. But it will be said that our experience can never extend to infinity and we are therefore not warranted in extending any principle of infinity. I do not desire to enter into the metaphysics of the question. I will only say that what holds good within the limits of our experience may be fairly assumed to hold good beyond these limits until we see some reason to the contrary. And # By mistake this cut was not engraved. The reader can easily supply it by reference to the figure on page 336 with the following suggestions: AB is the same in both figures, if A and B be placed respectively a little higher and lower than at the points of intersection of the lines. In the wanting cut CF is the same as AC, CJ as AE, and EH as BD. Then draw a parallel line, DG, midway between CF and EH and the new figure is complete.-Ed. I think if Euclidean geometry did not hold good with respect to the most distant stars that we have tried, its failure ought to be brought to light by our researches on parallax. If the three angles of a triangle with very long sides fell short of 180° we should probably find a considerable parallax for any very remote star; while if it exceeded 180° we ought to discover many more negative parallaxes-perhaps of considerable amount-than we have done. Our researches are as far as I know grounded on the assumption that the three angles of the triangle (whose base consists of the line going two points on the earth's orbit and whose base angles are deduced from observation) are equal to 180°. Nor is this the only instance in which such an assumption is made; but it is not necessary to enter into details. Although I am a firm believer in Euclidean geometery I am quite prepared to accept any evidence derived from observations. on stars presumably very distant which seems to be irreconcilable with it. If Euclid be wrong the stars alone can prove him to be so and I think that our present knowledge of the stars has progressed far enough to throw considerable light on the subject. I should therefore be glad to hear from some non-Euclidean what are the observed facts of astronomy which are relied on as inconsistent with Euclidean geometry-a kind of geometry whose validity as regards terrestrial measurements seems to be established by ample experience. Let me add however that in my opinion Euclid's geometry is not a mere logical deduction from his assumptions. He appeals at every step to our knowledge of space but this knowledge is so simple and universal that the reader is apt to overlook the fact that Euclid is always dealing with realities and not with mere argument. If you cannot draw the figure or draw it wrongly you cannot follow him. If when you were asked to bisect the base of a triangle by a right line from the vertex, you drew the line outside of the triangle and made the connection below the base, the conclusion will probably not follow. This, it may be said, would not be a right line. But can you prove this from Euclid's definition of a right line? If there were such a being as a man who had no idea of space he would find himself utterly unable to deduce any conclusion from Euclid's definitions and axiEuclid no doubt assumes that the properties of space are the same everywhere and at all times; but his constant appeal is, "Draw your figure. Test what I say by the space which you know and experience. You will not see the force of my reasonings until you do that." oms. I have treated Euclid's axiom as an assumption that the three angles of every plane triangle are equal to two right angles. If that fact could be proved otherwise the axiom might be dispensed with. Euclid puts it to no other use. Various forms of the axiom have been given with the view of making it more acceptable. Its acceptance I think does not depend on the terms but on the the realization of their import by the hearer. A Let the sum of the angles BAC, ABD be equal to two right angles (180°). It is susC ceptible of proof that the right lines AC and BD E cannot meet. Draw the B D right line AE. Can you believe that however both lines may be pro duced AE will never meet BD? The difficulty of belief does not arise from the terms employed but from gazing at or contemplating the figure. SPECTROSCOPIC NOTES. The entire path of totality of the eclipse of May 28 has been favored with a perfect sky. European observers in Portugal, Spain, and Algiers report clear weather. In this country all along the central line weather conditions were magnificent; the good fortune of which is accentuated by the fact of partial cloudiness as near to the path as New York. The instrumental equipment, which was elaborate, was installed chiefly at stations in North Carolina, Georgia and Alabama. Wadesboro, N. C., Pinehurst, N. C., Thomaston, Ga., Barnesville, Ga., Greenville, Ala., and Fort Deposit, Ala., might perhaps be mentioned as favorite points. Of the congregations of visiting sight-seers that at Norfolk, Va., was undoubtedly the most prominent. The amount of apparatus was large, and its variety considerable, including instruments adapted to almost every possible line of research, visual, photographic, spectroscopic, bolometric. Of the spectroscopic results a few only have been reported through the medium of the daily press; the great mass being largely spectrographic, must await development of plates and subsequent publication in the scientific journals. M. Paulsen (Comptes Rendus, March 5; Nature, April 26) during brilliant auroral displays in Iceland from Dec. 31 to Jan. 25 has succeeded in photographing a number of new bright lines in the violet and ultra-violet of the spectrum of the aurora. The statement has been in general circulation that spectroscopic evidence of the rotation of Venus has at last been secured. The accuracy of measurement of rotation in the line of sight has for some time been adequate to the detection of the rotation of Venus if the period is as short as a day; in fact the persistent failure to find evidence of rotation has perhaps come to be regarded as evidence on the side of those who contend for a period of 225 days. If the planet rotates in a day the velocity would be about a third of a mile per second. The effective velocity at time of superior conjunction would be two-thirds of a mile per second, or, as near conjunction as observations could be taken, perhaps half a mile per second; a quantity at present measurable. For the present the announcement of Belopolsky's results goes no further than the bare statement that the period of the planet's rotation is found to be short. In Astronomische Nachrichten No. 3633 Mr. Epsin publishes a further liberal list of stars with remarkable spectra. The stars are with one exception fainter than the seventh magnitude, and most of them are more or less striking examples of type III. A large majority of the stars of the present list are situated in the Milky Way near Cygnus. At the meeting of the Royal Astronomical Society, March 9, Mr. Shackleton (Monthly Notices, March; Observatory, April) read a paper describing his apparatus for examining the distribution of matter in the sun's corona. A photograph was exhibited, taken near the middle of totality in the eclipse of 1896, showing the continuous spectrum to a distance of 15′ from the moon's limb, while the coronium line at λ 5303 reached a height of only 5'. Mr. Shackleton described a combination of color screens admitting a narrow band of green light in the neghborhood of A 5303 and suppressing the rest of the spectrum. If coronium is confined to the inner corona, photographs taken through such a screen chiefly by the coronium radiation should be less extended and less irregular in form than ordinary photographs in which the entire continuous spectrum of the outer corona is active. Prof. FitzGerald contributes to Nature of May 3 a short note in support of the idea that the sun's corona is of the nature of a solar aurora. He cites, to explain the absence of a dark line of coronium in the sun's spectrum, recent experiments by Herr Cantor which seems to show that when the radiation of a gas is due to electric discharge there is no corresponding absorption. Mr. Newall contributes to the Monthly Notices of the Royal Astronomical Society for March some results of his study of the spectroscopic binary Capella. His former period of 104 days is confirmed. The observations show that the two components of different types of spectrum are nearly equal in mass, and are not very different in brightness. The radius of the relative orbit if it is seen edgewise is about 52,000,000 miles, or if seen inclined at an angle of 60° to the line of visior. 104,000,000. Assuming as accurate Elkin's rather uncertain value of 0".08 for the parallax, the actual brightness is about 500 times that of the Sun; for the orbit assumed to be seen edgewise the angular separation is 0".04 and the mass 1.7 that of the Sun; for the orbit assumed to be seen under an angle of 60° the angular separation is 0".08 and the mass 14 that of the Sun. |