September 6, 1889.] 



SCIENCE. 



167 



nomic Section on the " Economic and Sociologic Relations of the 

 Canadian States and the United States, prospectively considered," 

 roused considerable criticism. The meeting adjourned to meet 

 next year on the third Wednesday in August at Indianapolis. The 

 officers of the meeting will be as follows : president. Professor 

 George L. Goodale, Harvard University ; vice-presidents, A, Math- 

 ematics and Astronomy, S. C. Chandler, Cambridge, Mass. ; B, 

 Physics, Cleveland Abbe, Washington ; C, Chemistry, R. B. 

 Warder, Washington ; D, Mechanical Science and Engineering, 

 James E. Denton, Hoboken, N. J. ; E, Geology and Geography, 

 John C. Branner, Little Rock, Ark. ; F, Biology, C. S. Minot, Bos- 

 ton, Mass. ; H, Anthropology, Frank Baker, Washington ; I, Eco- 

 nomic Science and Statistics, J. Richards Dodge, Washington ; 

 permanent secretary, F. W. Putnam, Cambridge, Mass., office, 

 Salem, Mass. ; general secretary, H. Carrington Bolton, of New 

 York ; secretary of the council, James Loudon, Toronto ; secreta- 

 ries of the sections. A, Wooster A. Beman, Ann Arbor, Mich. ; B, 

 W. Le Conte Stevens, Brooklyn, N. Y. ; C, W. A. Noyes, Terre 

 Haute, Ind. ; D, M. E. Cooley, Ann Arbor, Mich. ; E, Samuel Cal- 

 vin, Iowa City, Iowa ; E, John M. Coulter, Crawfordsville, Ind. ; 

 H, Joseph Jastrow, Madison, Wis. ; I, S. Dana Horton, Pomeroy, 

 Ohio ; treasurer, William Lilly, of Mauch Chunk, Penn. ; auditors, 

 Henry Wheatland, Salem, Mass.; Thomas Meehan, Philadelphia. 



THE MATHEMATICAL THEORIES OF THE EARTH.' 



The name of this section, which, by your courtesy, it is my 

 duty to address to-day, implies a community of interest among 

 astronomers and mathematicians. This community of interest is 

 not difficult to explain. We can of course imagine a considerable 

 body of astronomical facts quite independent of mathematics. 

 We can also imagine a much larger body of mathematical facts 

 quite independent of and isolated from astronomy. But we never 

 think of astronomy in the large sense without recognizing its de- 

 pendence on mathematics, and we never think of mathematics as 

 a whole without considering its capital applications in astronomy. 



Of all the subjects and objects of common interest to us the 

 earth will easily rank first. The earth furnishes us with a stable 

 foundation for instrumental work and a fixed line of reference, 

 whereby it is possible to make out the orderly arrangement and 

 procession of our solar system and to gain some inkling of other 

 systems which lie within telescopic range. The earth furnishes us 

 with a most attractive store of real problems : its shape, its size, 

 its mass, its precession and nutation, its internal heat, its earth- 

 quakes and volcanoes, and its origin and destiny, are to be classed 

 with the leading questions for astronomical and mathematical re- 

 search. We must of course recognize the claims of our friends the 

 geologists to that indefinable something called the earth's crust, 

 but, considered in its entirety and in its relations to similar bodies 

 of the universe, the earth has long been the special province of as- 

 tronomers and mathematicians. Since the times of Galileo and 

 Kepler and Copernicus it has supplied a perennial stimulus to ob- 

 servation and investigation, and it promises to tax the resources of 

 the ablest observers and analysts for some centuries to come. 

 The mere mention of the names of Newton, Bradley, d'Alembert, 

 Laplace, Fourier, Gauss, and Bessel calls to mind not only a long 

 list of inventions and discoveries, but the most important parts of 

 mathematical literature. In its dynamical and physical aspects the 

 earth was to them the principal object of research, and the thor- 

 oughness and completeness of their contributions toward an expla- 

 nation of the " system of the world " are still a source of wonder 

 and admiration to all who take the trouble to examine their works. 



A detailed discussion of the known properties of the earth and 

 of the hypotheses concerning the unknown properties, is no fit task 

 for a summer afternoon : the intricacies and delicacies of the sub- 

 ject are suitable only for another season and a special audience. 

 But it has seemed that a somewhat popular review of the state of 

 our mathematical knowledge of the earth might not be without in- 



* Address before the Section of Mathematics and Astronomy of the American As- 

 sociation for the Advancement of Science, at Toronto, Ont., Aug. 28-Sept. 3, by R. 

 S. Woodward, vice-president of the s 



terest to those already familiar with the complex details, and might 

 also help to increase that general interest in science, the promo- 

 tion of which is one of the most important functions of this asso- 

 ciation. 



As we look back through the light of modern analysis, it seems 

 strange that the successors of Newton, who took up the problem 

 of the shape of the earth, should have divided into hostile camps 

 over the question whether our planet is elongated or flattened at 

 the poles. They agreed in the opinion that the earth is a spheroid, 

 but they debated, investigated, and observed for nearly half a cen- 

 tury before deciding that the spheroid is oblate rather than oblong. 

 This was a critical question, and its decision marks perhaps the 

 most important epoch in the history of the figure of the earth. 

 The Newtonian view of the oblate form found its ablest supporters 

 in Huyghens, Maupertuis, and Clairaut, while the erroneous view 

 was maintained with great vigor by the justly distinguished Cas- 

 sinian school of astronomers. Unfortunately for the Cassinians, 

 defective measures of a meridional arc in France gave color to the 

 false theory and furnished one of the most conspicuous instances 

 of the deterring effect of an incorrect observation. As you well 

 know, the point was definitely settled by Maupertuis's measure- 

 ment of the Lapland arc. For this achievement his name has be- 

 come famous in literature as well as in science, for his friend Vol- 

 taire congratulated him on having " flattened the poles and the 

 Cassinis," and Carlyle has honored him with the title of " Earth- 

 flattener." 



Since the settlement of the question of the form, progress to- 

 wards a knowledge of the size of the earth has been consistent and 

 steady, until now it may be said that there are few objects with 

 which we have to deal whose dimensions are so well known as the 

 dimensions of the earth. But this is a popular statement, and Uke 

 most such, needs to be explained in order not to be misunderstood. 

 Both the size and shape of the earth are defined by the lengths of 

 its equatorial and polar axes ; and, knowing the fact of the oblate 

 spheroidal (orm, the lengths of the axes may be found within nar- 

 row limits from simple measurements conducted on the surface, 

 quite independently of any knowledge of the interior constitution 

 of the earth. It is evident in fact, without recourse to mathemati- 

 cal details, that the length of any arc, as a degree of latitude or 

 longitude, on the earth's surface, muse depend on the lengths of 

 those axes. Conversely, it is plain that the measurement of such 

 an arc on the surface and the determination of its geographical 

 position, constitute an indirect measurement of the axes. Hence 

 It has happened that scientific as distinguished from practical 

 geodesy has been concerned chiefly with such linear and astro- 

 nomical measurements, and the zeal with which this work has been 

 pursued is attested by triangulations on every continent. 



Passing over the earlier determinations as of historical interest 

 only, all of the really trustworthy approximations to the lengths of 

 the axes have been made within the half century just passed. The 

 first to appear of these approximations were the well-founded values 

 of Airy, published in 1830. These, however, were almost wholly 

 overshadowed and supplanted eleven years later by the values of 

 Bessel, whose spheroid came to occupy a most conspicuous place 

 in geodesy for more than a quarter of a century. Knowing as we 

 now do that Bessel's values were considerably in error, it seems 

 not a Uttle remarkable that they should have been so long ac- 

 cepted without serious question. One obvious reason is found in 

 the fact that a consideraole lapse of time was essential for the ac- 

 cumulation of new data, but two other possible reasons of a differ- 

 ent character are wortay of notice, because they are interesting and 

 instructive whether specially applied to this particular case or not. 

 It seems not improbable that the close agreement of the values of 

 Airy and Bessel, computed independently and by different methods, 



— the greatest discrepancy being about one hundred and fifty feet, 



— may have been incautiously interpreted as a confirmation of 

 Bessel s dimensions, and hence led to their too ready adoption. It 

 seems also not improbable that the weight of Bessel's great name 

 may have been too closely associated in the minds of his followers 

 witti the weight of his observations and results. The sanction of 

 eminent authority, especially if there is added to it the stamp of an 

 official seal, is sometimes a serious obstacle to real progress. We 

 cannot do less than accord to Bessel the first place among the 



