QUADRATURE 



QUADRATURES 



515 



It has lone been known that the arithmetical 

 solution of the problem is impossible, for it has 

 been proved that the quantity IT is incommensur- 

 able. And proofs have been advanced that the 

 geometrical quadrature is also impossible ; but 

 these proofs are by no means simple, and do not 

 always convince tnose who are able to judge of 

 their accuracy. Still, apart from such proofs, the 

 mere consideration of the fact that (discounting 

 incapable workers ) the question has been fruitlessly 

 attacked by the ablest mathematicians of past 

 centuries should be sufficient to deter any reason- 

 able person from engaging in the quest : for it 

 follows that the probability of a solution being 

 possible is excessively small too small to justify 

 the staking of a man's sanity, or at least the useful- 

 ness of his life, upon the result. Any mathema- 

 tician who now considers the question seeks not 

 for a solution, but for a simple and convincing 

 proof that a solution is impossible. ( It must be 

 remembered that a 'geometrical' solution means 

 a solution which involves no more postulates than 

 those of Euclid. ) 



James Gregory, in 1668, gave a proof of the 

 impossibility of the geometrical quadrature which 

 Huygens, although he at first objected to it, 

 finally admitted in so far as it applied to any 

 sector of a circle. Newton also gave a proof of this 

 limited problem, but his proof is not conclusive. 



Archimedes was the first to give a practical 

 measurement of the quantity r. By a considera- 

 tion of the inscribed and escribed 96-gons he 

 proved that it lies between 3}g and 3J?. This 

 result is correct only to the second decimal figure. 

 Two Hindu measurements are 3-1416 and 3'1623. 

 Ptolemy gives 3-141552. A great improvement on 

 previous results was made by Peter Metius in the 

 16th century. His result was correct to the sixth 

 decimal place inclusive ; but its correctness was 

 accidental, for he gave two fractions between 

 which the resnlt lay and took the arithmetical 

 means of the numerators and the denominators in 

 order to obtain his linal numerator and denomin- 

 ator a totally unwarranted method. Vieta gave 

 the result correct to the ninth decimal place inclu- 

 sive ; Adrianus Komanns gave it correct to the fif- 

 teenth ; and Van Ceulen gave it to the thirty-sixth. 

 Snell introduced considerable improvements in the 

 method, and gave 55 decimal figures. Abraham 

 Sharp gave 75, Machin 100, De Lagny 128, Vega 

 140. The latter result is only correct to 136 places. 

 Montucla cites an Oxford manuscript in which the 

 result (given to 154 places) is correct to 152 places. 

 In 1846 I >.-ise gave a result with 200 decimals, and, 

 in the following year, Clausen gave 2oO. In 1851 

 Shanks gave 315, which were extended by Ruther- 

 ford to 350 ; and, shortly afterwards, Shanks gave 

 527, which he extended to 607. An interesting 

 experimental method was adopted by K. A. Smith. 

 He tossed a thin rod upon a uniformly planked 

 floor, the length of the rod being three-fifths of the 

 breadth of a plank. If I be the length of the rod, 

 while b is the breadth of a plank, the probability 

 of the rod intersecting a seam is 2//ir6. From the 

 result of 3204 tosses, he found w = 3-1412. The 

 true value to 20 places is 3-14159265358979323846. 



Any one who is desirous of a more detailed 

 historical account may consult I)e Morgan's article 

 on the subject in his Budget of Paradoxes (1872). 



<fii:ulr:iliirrs. METHOD OF. This name is 

 applied to any arithmetical method of determining 

 the area of a curve. When the exact area is 

 known a square whose area is equal to it can be 

 found hence the term 'quadratures.' 



It has been shown, under the heading CALCULUS, 

 that the area of a curve whose equation is y = f(x) 

 is fydx, and can therefore be found when the 

 integral can be evaluated. Hence the approximate 



determination of the value of a definite integral is 

 obtainable by the method of quadratures. 



Let it be 'required to find the area bounded by a 

 portion of a curve, the ordinates at its extremities, 

 and the axis. The usual method of procedure is to 

 divide the portion of the axis which is included 

 between the two ordinates into a number of equal 

 parts, and to erect ordiuates at the points so 

 obtained. The area is approximately equal to 

 the product of one of the given equal parts into 

 half of the sum of the two extreme ordinates 

 together with the sum of all the intermediate 

 ordinates. To obtain, a very accurate result by 

 this process the number of equidistant ordinates 

 must be so great that the portions of the curve 

 which are intercepted by successive ordiuates are 

 very nearly straight. 



A better method, due to Simpson, consists in 

 drawing, through the first, second, and third points 

 obtained as above on the curve, a parabola whose 

 axis is parallel to the ordinates, an a repeating this 

 process with the third, fourth, and fifth points, and 

 so on the points being chosen so that the total 

 number of points is even. The area of the given 

 curve will oe approximately equal to the sum of 

 the areas of the various portions of the parabolas 

 included between successive ordinates when these 

 ordinates are sufficiently close together. It is 

 therefore approximately equal to one-third of the 

 product of one of the given equal portions of the 

 axis into the sum of the extreme ordinates together 

 with twice the sum of all the odd intermediate 

 ordinates and four times the sum of all the even 

 intermediate ordinates. 



When the successive equidistant ordinates are 

 very close together, the area is approximately 

 equal to the product of the common intercept on 

 the axis between successive ordinates into the sum 

 of all the ordinates. The labour involved in the 

 estimation of an area by this process would be 

 fatal to its employment unless the number of 

 ordinates was small. But, if the ordinates were 

 few in number, considerable error would in general 

 result unless a correction could be applied. This 

 method is adopted in that process which is known 

 as the method of quadratures par excellence, and 

 which is as follows : Let y , y v . . . y, be the 

 several equidistant ordinates, and let a be the 

 intercept on the axis letween y and y,,. Also let 

 i be the sum above referred to; and let Ay = y l 

 -y, Ay, = y.j - y,, &c. ; A a y u 

 A 2 y, = Ay t - Ay,, &c. ; and so on. 

 the whole area is ( not a, but ) 



3 A 



- A ^ 



+ A a y) - 



Ay, - Ay , 

 The value of 



AV* - A'yo) 

 '.( A y^ - A 5 y ), &c. 



It will not in general be necessary to proceed 

 beyond the fifth diti'erence. As an example we 

 shall find the area of the curve y = x 3 between the 

 limits x 10 and x = 15. In this case all differences 

 beyond the third vanish, and a/n = 0'5 if we make 

 eleven ordinates in all. The following table repre- 

 sents the results : 



