144: ANNUAL OF SCIENTIFIC DISCOVERY. 



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 determination of a square 

 equal to the circle, using only Euclid's allowance of means ; that is, using 

 only the straight lino and circle as in Euclid's first three postulates. On this 

 matter, James Gregory, in 1668, published an asserted demonstration of the 

 impossibility of the geometrical quadrature. The matter is so difficult, and 

 proofs of a negative so slippery, that mathematicians are rather shy of pro- 

 nouncing 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 edition, 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 consideration 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 reason- 

 ing, different from Gregory's, which render it in the highest degree improb- 

 able, which arc 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 succeeded, is unsafe ; far 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 insolu- 

 ble. By hypothesis, we are to use no means except those which have been 

 used for two thousand 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 repre- 

 senting 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 considerations which have so much 

 force with those who are used to them are in favor of the solution being im- 

 possible. A person used to algebraic gcometiy cannot conceive how, by in- 

 tersections 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 wanting 

 only pains and patience to carry the approximation to any desired extent. 

 The problem of the perpetual motion is a physical absurdity. The arith- 

 metical quadrature of the circle has been proved impossible in finite terms, 

 but 607 decimal places of the interminable scries have been found, and 441 

 of them verified. Of the geometrical quadrature an asserted proof of im- 

 possibility exists, which no one who has read it ventures to gainsay, but in 

 favor of which no one speaks very positively. The trisection of the angle 

 has no alleged proof of its impossibility. But were this the proper place, an 



