142 ANNUAL OF SCIENTIFIC DISCOVERY. 



all deferential equations should be soluble in what are called finite terms, that 

 is by a definite number of algebraical, etc., terms consisting of our usual 

 modes of expression. Mathematicians had not then opened their eyes to the 

 fact that there exists an unlimited number of modes of expression of which 

 those we employ cannot give an idea, except by interminable series. Ac- 

 cordingly, they considered the problem of three bodies unsolved so long as 

 it was necessary to have recourse to these interminable series. But is this 

 problem unsolved, in any other sense than this, that the nature of things has 

 not listened to human dictation on matters which humanity knew nothing 

 about? Do we not find the moon's place within a fraction of a second of 

 time, by the existing solution ? And did not Adams and Leverricr even 

 solve the inverse problem, Given the effect produced upon a known planet 

 by an unknown planet, to discover the place of the unknown planet ? There 

 are hundreds of problems, in pure and mixed mathematics both, which are 

 treated only by interminable series, and which no one ever complained of as 

 not being solved. The difference is this : we speak of these problems in the 

 language of the newer day ; we speak of the problem of the three bodies 

 after the tradition of an older day. 



It is not practicable, that is, it has not been found practicable, to prove the 

 impossibility of solving the problem of three bodies without interminable 

 series. But a long chain of cogent analogies convinces every one who has 

 gone through them, with full moral evidence, that the finite terms must be 

 terms of a kind of which we have at present no conception. 



2. The Perpetual motion. This is a problem of a very different kind. 

 The purse of Fortunatus, which could always drop a penny out, though 

 never a penny was put in, is a problem of the same kind. He who can con- 

 struct this purse may construct a perpetual motion ; in this way. Let him 

 hang the purse upside down, and with the stream of pence which will flow 

 out, let him buy a strong steam engine, and pay for keeping it at work day 

 and night. Have a new steam-engine ready to be set in motion by tho old 

 one at its last gasp, and so on to all eternity. A perpetual motion demands 

 of the nature of things a machine which shall always communicate momen- 

 tum in the doing of some work, without ever being fed with any means-col- 

 lecting momentum. It could be compassed, in a certain way, that is, by 

 retaining the work done to do more work, which again should be more, and 

 soon, if friction and other resistances could be abolished, and nothing 

 thrown away. In this way the fall of a ton of water from a reservoir might 

 be employed in pumping up as much water into another reservoir, which, 

 when landed, if it be laAvful to say so of water, might, by its subseqiient fall, 

 pump up an equal quantity into the original reservoir, and so on, backAvarcls 

 and forwards, in secula seculorum. But not a drop must be wasted, whether 

 by adhesion to the reservoir, by evaporation, by splashing, or in any way 

 whatever. Every drop that falls down must be made to raise another drop 

 to the same height. So long as the sockets have friction, or the air resists, 

 this is impossible. In fact, matter, with respect to momentum, has the 

 known qualities of a basket with respect to eggs, butter, garden-stuff, etc. 

 No more can come out than was put in ; and every quantity taken out re- 



