LIFE AND WORKS OF KEPLER. 101 



your treasurer to deliver to your geueral the sums necessary for the 

 levy of new troops." 



In couiuiencing the study of the movements of Mars, it was incumbent 

 on Kepler to ascertain with precision the time of its revolution, which 

 was not unknown to Tycho, nor even to Ptolemy, who had calculated it 

 with nearly equal exactuess. It is a problem, in fact, which, notwith- 

 standing- its apparent diflticalties, is of easy solution. The imaginary 

 right line, called the radius vector, which counects the fixed center of 

 the sun with the movable center of the planet, may be compared to the 

 hand of a clock, and the time occupied in traversing its vast dial is the 

 time of the revolution of jNIars ; the radius vector which unites the earth 

 with the sun may be regarded as a -shorter hand than the preceding 

 and as turning in the same direction. The movement of the latter is 

 well known; it makes its circuit in a year. Suppose, now, though it 

 be not absolutely exact, that the planes of the two orbits coincide ; in 

 other terms that the two hands, of unequal length, move on the same dial- 

 plate. Placed as we are at the extremity of the smaller, it is easy for 

 us to note its coincidences with the greater, and the astronomers who 

 attentively observe the sun and the planet Mars will be able to say afe 

 what moment we are on the line which unites them. It has been long 

 known that these oppositions of Mars and the sun, or, what amounts to 

 the same thing, the coincidences of the two hands of the dial, take place 

 on a mean every 795 days. The longer hand, therefore, makes in 795 

 days one circuit less than the shorter ; and as the movement of the latter 

 is known to us, the tyro in astronomy can deduce therefrom the move- 

 ment, assumed to be uniform ; that is to say, the mean movement of the 

 other. It is thus that the period of the revolution of Mars has been 

 found equal to G87 days. 



This result being well known to Kepler, he conceived the idea of col- 

 lating, in the observations of Tycho, those which differed precisely by 

 that number of days, and for which, consequently, Mars, after having 

 accomplished a circuit, had returned to the same point of its course. 

 He thus very ingeniously eluded the difficulty, apparently insurmount- 

 able, which results from its continual displacement in space. The two 

 l)Ositions of the earth in its orbit being known through the previous 

 study which had been made of its movement, the line which unites them 

 becomes the base, at the two extremities of which the inquirer is con- 

 sidered as placed for the observation of a planet, which, having returned 

 to the same position, may be regarded as motionless. One of the posi- 

 tions of Mars will thus be found with the date of two epochs, separated 

 by an interval of (387 days, on which it has arriv^ed at that place. By 

 interposing other observations separated from the first by a period of 

 two or three revolutions of the planet, the same result will be obtained, 

 a result which furnishes a means of verifying the calculations, and at 

 the same time, what is still more valuable, a confirmation of the hypoth- 

 esis adopted for the law of the movement of the earth. 



Encouraged by this first success, Kepler recommenced the operation 

 a great number of times, following the planet step by step, in order, so 

 to say, to stake out its course through space ; but how many points are 

 needed to determine the geometric nature of a curve ? Eigorous geom- 

 etry answers that, however great the number, it will not suffice, and 

 that by any given j)oints an infinite number of distinct curves of very 

 different properties may always be made to pass ; it is for this reason 

 that so many tables admirably precise obtained by physicists have never 

 been found susceptible, notwithstanding their efforts, of being converted 

 into mathematical laws. The uncertainty and iucomxietence of science 



