274 



NOTES AND QUERIES. 



[2»-i S. No 66., April 4. '57. 



of a periodical produced an article which, I be- 

 lieve, destroyed the concern. The plan was to 

 put a drum or broad wheel with one vertical half 

 m mercury and the other in vacuum. This in- 

 strument, the most unlucky drum since Parolles, 

 feeling the balance of its two halves very unsatis- 

 factory, was to go round and round in search of 

 an easy position, for ever and ever, working away 

 all the time, — I mean all the eternity — at lace- 

 making, or water-pumping, or any other useful 

 employment. People were told that if they would 

 sell their steam-engines for old iron, they might 

 buy new machines with the money, which would 

 work as long as they held together without cost- 

 ing a farthing for fuel. Certainly, had the scheme 

 been proposed to me, I should have declined to 

 join until I had derived assurance from seeing the 

 donkey who originated it turned into a head-over- 

 heels perpetual motion by tying a heavy weight 

 to his tail and an exhausted receiver to his nose. 



3. Quadrature of the circle. The arithmetical 

 quadrature involves the determination of the cir- 

 cumference by a definite arithmetical multiplier, 

 which shall be perfectly accurate. Lambert 

 proved that the multiplier must be an intermin- 

 able decimal fraction : and the proof may be found 

 in Legendre's geometry, and in Brewster's trans- 

 lation of that work. The arithmeticians have 

 given plenty of approximate multipliers. The 

 last one, and the most accurate of all, was pub- 

 lished a few years ago by Mr. W. Shanks, of 

 Houghton-le-Spring, a calculator to whom multi- 

 plication is no vexation, &c. He published the 

 requisite multiplier (which mathematicians denote 

 by 7r) to six hundred and seven decimal places, of 

 which 441 were verified by Dr. Rutherford. To 

 give an idea of the power of this multiplier, we 

 must try to master such a supposition as the fol- 

 lowing. 



There are living things on our globe so small that, 

 if due proportion were observed, the corpuscles of 

 their blood would be no more than a millionth of 

 an inch in diameter. Suppose another globe like 

 ours, but so much larger that our great globe 

 itself is but fit to be a corpuscle in the blood of 

 one of its animalcules : and call this the first globe 

 above us. Let there be another globe so large 

 that this first globe above us is but a corpuscle 

 in the animalcule of that globe : and call this 

 the second globe above us. Go on in this way 

 till we come to the twentieth globe above us. 

 Next, let the minute corpuscle on our globe be 

 another globe like ours, with every thing in 

 proportion ; and call this the first globe below 

 us. Take a blood-corpuscle from the animalcule 

 of that globe, and make it the second globe below 

 us : and so on down to the twentieth globe below 

 us. Then if the inhabitants of the twentieth globe 

 above us were to calculate the circumference of 

 their globe from its diameter by the 607 decimals, 



their error of length could not be made visible to 

 the inhabitants of the twentieth globe below us, 

 unless their microscopes were relatively very much 

 more powerful than ours. 



By the geometrical quadrature is meant the de- 

 termination of a square equal to the circle, using 

 only Euclid's allowance of means ; that is, using 

 only the straight line and circle as in Euclid's 

 first three postulates. On this matter James 

 Gregory, in 1668, published an asserted demon- 

 stration of the impossibility of the geometrical 

 quadrature. The matter is so difiicult, and proofs 

 of a negative so slippery, that mathematicians are 

 rather shy of pronouncing positive opinions. 

 Montucla, in the first edition of the work pre- 

 sently mentioned, only ventured to say that it 

 was very like demonstration. In the second edi- 

 tion, after further reflection, he gave his opinion 

 that the point was demonstrated. I read James 

 Gregory's tract many years ago, and left off" with 

 an impression that probably more attentive con- 

 sideration would compel me to agree with its 

 author. But he would be a bold man who would 

 be very positive on the point : even though there 

 are trains of reasoning, different from Gregory's, 

 which render it in the highest degree improbable, 

 which are in fact all but demonstration themselves, 

 that the geometrical quadrature is impossible. 



To say that a given problem cannot be solved, 

 because two thousand years of trial have not suc- 

 ceeded, is unsafe : for more powerful means may 

 be invented. But when the question is to solve a 

 problem with certain given means and no others, 

 it is not so unsafe to affirm that the problem is 

 insoluble. By hypothesis, we are to use no means 

 except those which have been used for two thou- 

 sand years : it becomes exceedingly probable that 

 all which those means can do has been done, in a 

 question which has been tried by hundreds of men 

 of genius, patience, and proved success in other 

 things. 



4. Trisection of the Angle. — The question is to 

 cut any given angle into three equal parts, with 

 no more assistance than is conceded in Euclid's 

 first three postulates. It is well known that this 

 problem depends upon representing geometrically 

 the three roots of a cubic equation which has all 

 its roots real : whoever can do either, can do the 

 other. Now the geometrical solution, as the word 

 geometrical is understood, of a cubic equation, 

 has never been attained : and all the a priori con- 

 siderations which have so much force with those 

 who are used to them, are in favour of the solu- 

 tion being impossible. A person used to algebraic 

 geometry cannot conceive how, by intersections 

 of circles and straight lines, a problem should be 

 solved which has three answers, and three only. 



To sum up the whole. The problem of the three 

 bodies has such solution as hundreds of other 

 problems have; approximate in character, but 



