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SCIENCE 



[N. S. Vol. XXXIII. No. 845 



world to prescribe the boundaries for their 

 activities, nor do they in turn feel quali- 

 fied or even inclined to impose any limita- 

 tions upon the mathematicians. They 

 could not even say what kinds of mathe- 

 matics will be of use to themselves or to 

 other branches of physical science. To 

 take an example old enough to be under- 

 stood in correct perspective, the interval 

 between the discovery of the properties of 

 the conic sections by Menjechmus and their 

 first practical use by Kepler was 2,000 

 years, or more than nine times that which 

 separates us from Newton. Astronomers 

 will admit, then, that if the sole purpose of 

 mathematics were to serve the other sci- 

 ences, it would not be safe to circumscribe 

 it by any boundaries. And most of them, 

 I think, will go much further and join me 

 in the sentiment that mathematics, alto- 

 gether apart from its uses in other subjects, 

 has a right to exist ; that it is a part of the 

 universe of ideas which to a thinking being 

 is no less real and important than the 

 physical universe; that its proportions and 

 its symmetries which tind perfect expres- 

 sion in its wonderful symbolism are, in 

 satisfying the esthetic tastes, on a level 

 with the fine arts; and that the process of 

 drawing its conclusions calls for an exercise 

 of the best and highest faculties we pos- 

 sess. If we were required to describe the 

 proper field of mathematics we might say 

 simply that it includes at least all that 

 which all mathematicians together claim 

 belongs to it. 



Having admitted the breadth of mathe- 

 matics, we have to consider what part of it 

 has had at least its initial inspiration in 

 astronomy. It might, perhaps, be argued 

 with a good deal of jvistice that all of 

 mathematics has originated directly or in- 

 directly in the experience of the human 

 race ; that our capacity for those particular 

 modes of thought which are essential to its 



development have evolved under the stimu- 

 lus of the physical world. It is significant, 

 at any rate, that there is such wonderful 

 harmony between the results obtained by 

 mathematical processes and our experi- 

 ences. But it is not the purpose here to 

 make any such claims, or to become in- 

 volved in the diificulties of metaphysical 

 discussions. No thesis has been laid down 

 which it is necessary to defend, and no 

 claim that astronomy has had an important 

 influence on mathematics will be filed ex- 

 cept where the evidence is perfectly clear 

 and conclusive. 



It was noted in the beginning that in 

 ancient times the astronomers were almost 

 invariably also mathematicians and re- 

 versely, and consequently that it is diffi- 

 cult to separate the two sciences of that 

 time so as to determine exactly the influ- 

 ence which each had on the other. But 

 there is one case in which the demands of 

 astronomical problems certainly stimulated 

 the development of a mathematical theory. 

 Trigonometry was invented by Hip- 

 parchus, who was the most eminent Greek 

 astronomer, both as a practical observer 

 and as a mathematician. He determined 

 the length of the year correctly to within 

 six minutes of its true value, the obliquity 

 of the ecliptic to within five minutes of are, 

 the annual precession of the equinoxes to 

 within nine seconds of arc, the distance of 

 the moon to within one per cent, of its 

 value, the mean motions of the sun, moon 

 and known planets, the changes in the 

 moon's orbit, he made a catalogue of the 

 fixed stars, etc. There is every reason to 

 believe that these astronomical problems 

 were those in which he was chiefly inter- 

 ested, and they made it necessary for him 

 to develop trigonometry, and especially 

 spherical trigonometry. His work was 

 completed by Gauss nearly 2,000 years 



