QUADRATURE OF THE CIRCLE.] MATHEMATICS. PL ANE GEOMETRY. 



577 



skied polygon (Proposition XV.) to that of fifteen sides 

 (Proposition XVI.), without anything being said as to 

 the intermediate polygons of 7, 9, 11, 13, and 14 sides 

 respectively . You now know that the omission arose from 

 his inability to inscribe any of these polygons in a circle. 



ON THE QUADRATURE OF THE CIRCXE. 



Intimately connected with the researches just adverted 

 to, is the problem of the quadrature of the circle, which, 

 like that of the trisection of an angle, has for ages occa- 

 sioned the fruitless expenditure of much valuable time 

 and thought. We shall endeavour to give you here some 

 notion of the meaning and object of this celebrated 

 problem, not only because it is a matter of such historical 

 interest, that you ought to know something about it ; but 

 because, moreover, in certain elementary writings on 

 the subject, the thing is put before the student in an 

 erroneous form ; and, consequently, a wrong impres- 

 sion as to the real character of the problem is con- 

 veyed. 



The problem of squaring the circle, as it is popularly 

 called, has a twofold meaning namely, the geome- 

 trical quadrature, and the numerical quadrature. In 

 the first of these senses the problem is to construct a 

 square that shall be equal in surface to a given circle ; 

 in the second, the problem is to express the numerical 

 measure of the surface of a circle when the measure or 

 length of its diameter is given in numbers. The former 

 of these is the more ancient form of the problem ; and 

 all that can be fairly said of it is as was said of the 

 truection of an angle that the solution has never been 

 effected ; a square equal to a circle has never yet been 

 constructed. We have no grounds for affirming that 

 this construction is impossible, for the equivalent square 

 exitts. You may readily satisfy yourself of this by the 

 following reflections : The square on the diameter of the 

 circle would be too great, and the square on the chord 

 from an extremity of the diameter, to cut off a fourth 

 part of the circumference, would be too small, since the 

 furmi-r square would be circumscribed about the circle, 

 and the latter inscribed in it ; the circle therefore is in 

 magnitude somewhere between the two. Conceive, now, 

 the smaller of these two squares to expand continuously, 

 still retaining its character as a square, till it arrives at 

 the larger square in magnitude ; then, as all inter- 

 mediate magnitudes are thus reached and passed 

 through, and as the circle is one of these intermediate 

 magnitudes, it necessarily follows that our expanding 

 square must, at a particular stage of its progress, have 

 exactly attained the magnitude of the circle ; so that if 

 its progress could be arrested at that stage, or to drop 

 this idea of progression if the individual square could 

 be isolated and exhibited, the problem of the geometrical 

 quadrature of the circle would be solved. It is plain, 

 therefore, that there is nothing visionary or absurd in 

 the search after this square, as if it were a thing that 

 had no existence ; although some very able geometers 

 have, strangely enough, condemned the inquiry on these 

 grounds. The only sound reasons for abandoning the 

 investigation are these two ; namely first, that the 

 problem has been earnestly and laboriously attempted, 

 by the profouudest geometers, for thousands of years, 

 and they have been obliged to abandon it in despair ; 

 and secondly, that the successful solution of it would be 

 "f no theoretical or practical value if furnished. As far 

 an utility is concerned, the other form of the problem of 

 the quadrature of the circle is by far the more important ; 

 that is, t<> discover the numerical measure of the surface 

 of a circle from the measured length of its diameter 

 being given. But, under this aspect of it, the accurate 

 solution of the problem is really impracticable ; it can be 

 proved to be so ; and the proof will be given in a sub- 

 sequent part of the present mathematical course! It 'is 

 just as impracticable as it is to assign accurately the 

 si|iiai---rot of 2 ; and, in fact, this square-root does re- 

 v enter into the approximative numerical process. 

 You will require to know something of Proportion, in 

 the sense in which the term is employed by Euclid, 

 before that process can be fully explained to you. This 

 VOL. i. 



subject, together with the sixth book of Euclid, forms 

 the object of the chapter next 



following ; and at the end of it No. of sides. Surface of pol. 

 you will find the principles upon 4 2 



which the approximative quad- 8 2'8284271 



rature of the circle depends, 16 3-0614674 



clearly exhibited ; and the mode 32 31214451 



of computation pointed out. 64 3-1365485 



The plan is to compute first the 128 31403311 



surface of the inscribed four- 256 3-1412772 



sided equilateral figure or square; 512 3'1415138 



then the inscribed eight-sided 1024 3 1415729 

 figure ; then the sixteen-sided 2018 3-1415877 

 figure ; and so on, till the inscribed 4096 3 1 415914 



polygon differs insensibly from 8192 3 1415923 



the circle. How the surfaces of 16384 31415925 

 these successive polygons are 32768 3-1415926 

 computed one after another, 



must be deferred for the present ; but some of the results 

 are exhibited in the margin, where the radius of the 

 circle, whose surface is approximated to, is regarded as 

 1 ; that is, 1 inch, 1 foot, 1 yard, or one any measure. 



It appears, from the above table, that the surface of an 

 inscribed regular polygon of 32768 sides is 31415926, 

 which is correct as far as the decimals extend ; this 

 number, therefore, may be taken for the numerical 

 measure of the circle itself ; for it is plain that a regular 

 polygon of so many thousand sides would be undistin- 

 guishable from, and therefore practically identical with, 

 the circle in which it is inscribed. But, by continuing 

 to double the number of sides of the polygon, the ap- 

 proximation to the circle may be pushed to any extent. 

 De Lagny computed the decimals true to 128 places ; and 

 eighty more have been recently added on by Dr. Ruther- 

 ford. It is observed by Montucla, that if we suppose a 

 circle whose diameter is a thousand million times the 

 distance of the sun from the earth, the approximative 

 measure of the circumference, as computed by De Lagny, 

 would differ from the true measure by a length less than 

 the thousand millionth part of the thickness of a hair. 



There is obviously no use, as far as practical purposes 

 are concerned, in extending the approximation to any- 

 thing like this extreme degree of nearness. The earlier 

 computers were no doubt induced to carry on the deci- 

 mals in the expectation that they would at length ter- 

 minate ; but, as already observed, and as will be hereafter 

 proved, the decimals will go on for ever. It is right to 

 infer, therefore, that the exact numerical measure of the 

 surface of a circle does not exist, though a geometrical 

 square equal to it does exist. The exact numerical value 

 of J2 does not exist ; yet ^/2 represents the diagonal 

 of a square whose side is 1, which diagonal is of course 

 an existent geometrical line. 



EXERCISES ON BOOKS I. IV, 



1. Prove that if a pair of opposite an 'les or., a, quad- 

 rilateral be equal to two right angles, a 'circle maybe 

 described about it. (Converse of Proposition XXII., 

 Book III.) 



2. If the diagonals of a quadrilateral divide one an- 

 other, so that the rectangle contained by the parts of the 

 one is equal to the rectangle contained by the parts of 

 the other, then a circle may be described about the quad- 

 rilateral : required the proof. 



3. If two opposite sides of a quadrilateral be prolonged 

 to meet, and if it be found that the rectangle contained 

 by one of *he lines thus produced, and the part produced, 

 are equal to the rectangle contained by the other line 

 and the part produced, then a circle maybe described 

 about the quadrilateral : required the demonstration. 



These two theorems are virtually the converses of Propositions 

 XXXV. and XXXVI. of Book III. 



4. If a circle be described about the square B E (see 

 the Diagram to Proposition XLVII., Book I.), its cir- 

 cumference shall pass through the point where AD, F C 

 intersect, and also through the point where A E, K li 

 intersect : required the proof. 



6. If an equilateral triangle be constructed on one side 



4 E 



