2nd s. No 66., April 4. '57.] 



NOTES AND QUERIES. 



273 



simple definite number or fraction, the earliest 

 object of our attention, was declared to be the 

 universal mode of expression. It was prescribed 

 to the circle that it should be, in circumference, 

 a definitely expressible derivation from the dia- 

 meter : it was demanded of the nature of things 

 that by cutting the circumference into a certain 

 number of equal parts, a certain number of those 

 parts should give the diameter ; and vice versa. 



In geometry, Euclid laid down, as his prescribed 

 instruments, the straight line and circle. Of all 

 the infinite number of lines which exist, he would 

 use none except the straight line and circle. It 

 was demanded of the nature of things that it 

 should be possible to construct a square equal to 

 a given circle, without the use of any curve ex- 

 cept the circle. 



The second demand was not quite so impudent 

 as the first. It was soon discovered and proved 

 that there is no square root to 2, as a definite frac- 

 tion of a unit. That is, there is nothing but an in- 

 terminable series of decimals, 1'4142135 ; 



by help of which we discover the square root of 

 fractions within any degree of nearness to 2 we 

 please. And yet, with such a result as this known 

 to all, it was thought the most reasonable thing in 

 the world to demand that the ratio of the circum- 

 ference to the diameter should be that of number 

 to number. 



I will now speak of the problems set forth in 

 the question. 



1. The three bodies. This is the problem of 

 determining the motion of a planet attracted, not 

 only by the sun, but by another planet. In the 

 early days of the integral calculus, it was de- 

 manded of the nature of things that all differen- 

 tial equations should be soluble in what are called 

 finite terms, that is by a definite number of alge- 

 braical, &c. 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 intermin- 

 able series. Accordingly, they considered the 

 problem of three bodies unsolved so long as it 

 was necessary to have recourse to these intermin- 

 able 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 Leverrier 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 hun- 

 dreds of problems, in pure and mixed mathema- 

 tics 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 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 in- 

 terminable 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 construct this purse may con- 

 struct 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 the 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 momentum in the doing 

 of some work, without ever being fed with any 

 means of collecting momentum. It could be 

 compassed, in a certain way, — that is, by re- 

 taining the work done to do more work, which 

 again should do more, and so on, — 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 lawful to say so of 

 water, might, by its subsequent fall, pump up an 

 equal quantity into the original reservoir, and so 

 on, backwards and forwards, in secula seculorum. 

 But not a drop must be wasted, whether by adhe- 

 sion 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, &c. No more can come out 

 than was put in ; and every quantity taken out 

 requires as much more to be put in before the 

 original state is restored. So soon as the law of 

 matter i^ as clearly known as the law of the bas- 

 ket, there is an end of looking for the perpetual 

 motion. 



That people do try after a perpetual motion to 

 this day is certain. A good many years ago a 

 perpetual motion company was in contemplation ; 

 and the promoters did me the unsolicited honour 

 of putting my name on the list of directors. For- 

 tunately the intention came round to me before 

 the list was circulated : and a word to the editor 



