BOOK VLSUEF ACE OF POLYGONS.] MATHEMATICS. PLANE GEOMETRY. 



591 



Again, the sectors BGC, CGK, KGL are equal; 

 for it is manifest that, if applied to one another, they 

 would coincide ; in like manner, the sectors E H F, 

 F H M, M H N, are also equal ; consequently, if in the 

 preceding demonstration "sectors" are substituted for 

 " angles," the conclusion will be that B C : E F : : 

 tor B G C : sector EHF. .'.in equal circles, <tc. 

 Q. E. D. 



We shall conclude this chapter with two supplementary 

 propositions of interest. 



1. The diagonal and side of a square are incom- 

 mensurable. 



Let B D be a square : the diagonal A C is incommen- 

 surable with its side A B. 



With C as centre, and C B as radius, describe the 

 semicircle F B E : then ABC being a right angle, A B 

 I touches the circle (16 III.); , 



AE-AF = AB 2 .-. (17 



AB : : AB : AF 



D r/ \ . (23 V.) the lines AE, AB are 



incommensurable, . . AC, A B, 

 are also incommensurable ; for if 

 these had a common measure, 

 that measure would likewise mea- 

 sure A C + AB, that is AE ; so that 

 BA E would have a common 



measure, which is shown to be impossible : . . the 

 diagonal and side of a square are incommensurable. 



2. The surface of a regular inscribed polygon, and that 

 of a similar circumscribed polygon being given, to find the 

 surfaces of regular inscribed and circumscribed polygons of 

 double the number of side*. 



Let ab be a side of an inscribed polygon : the touching 

 line aA, 6C will each be half a side of the similar cir- 

 cumscribed polygon, as it is evident from what has been 

 shown in reference to the inscribed and circumscribed 



pentagon in Proposi- 

 tions XI. and XII. of 

 Book IV., and from 

 what is said at the close 

 of Prop. XVI. Let 

 the surfaces of these 

 two polygons be given, 

 v^ The chords aM, 6M, 

 > drawn to the middle 

 of the arc oMi), will be 

 sides of an inscribed 

 polygon of double the 

 number of sides, and 

 the touching line, 

 BMC, will be a side 

 of a similar circum- 

 scribing polygon, as is evident. 



For brevity, let the surface of the inscribed polygon, 

 whose side is 06, be represented by p, and that of the 

 corresponding circumscribed polygon by P ; also let the 

 inscribed and circumscribed polygons of double the num- 

 ber of sides be denoted by p' and P 7 - 



It is evident that the space OaD, is the same part of p, 

 that OaA is of P, that OaM is of p', and that OaBM 

 is of I" ; for each of these spaces must be repeated 

 exactly the same number of times to complete the poly- 

 gon to which it belongs : consequently, since magnitudes 

 are as their like multiples or sub-multiples, whatever 

 proportions exist among these spaces, must also exist 

 among the polygons of which they are sub-multiples. 

 Now the right angled triangles ODa, OAa are similar, 

 .-. OD : Oa : : Oa : O A ; that is, OD : O M : : OM 

 : O A ; and since triangles of the same altitude are to 

 one another as their bases, and that the altitude aD 



is the same for the triangles ODa, OMa, OAa, it follows, 

 from the proportion just deduced, that O Da : O Ma 

 : : O Ma : O Aa ; that is, the numerical measure of the 

 surface of the triangle OMa is a mean proportional be- 

 tween the measures of ODa, OAa ; consequently the 

 surface of the polygon p' is a mean proportional between 

 the surfaces of p and P. 



Again, the right-angled triangles O Da, B M A, are 

 also similar, .-.OD:Oa::BM:BA; that is, O D 

 : OM : : oB : B A ; consequently, since the altitude 

 aD is the same for the triangles O Da, O Ma, and the 

 altitude Oa, the same for the triangles OaB, OB A, and 

 that triangles of the same altitude are to each other as 

 their bases, it follows that ODa : OMa : OaB : OB A ; 

 .-. (13 and 6 V.) ODa + OMa : 2ODa : OaB+OBA 

 : 2OaB; .'.p+p' : 2p : : P F 

 The two conclusions now obtained are sufficient to enable 

 us to compute the surfaces of inscribed and circum- 

 scribed regular polygons of 8, 16, 32, <fcc. sides, from 

 having the surfaces of the inscribed and circumscribed 

 squares already given. Thus, let the radius of the circle 

 be numerically represented by 1, then if the given in- 

 scribed and circumscribed polygons (p, P) be squares, the 

 side of the former will be J 2, and that of the latter 2 ; 

 and their surfaces will be 2 and 4 respectively ; and, 

 from what is proved above, the surface of the regular 

 eight-sided inscribed polygon (p') will be a mean between 

 the two squares p, P, . . p'= J 8 = 2-8284271. Again, 

 for the surface of the eight-sided circumscribed polygon 

 (P") the proportion p -}-p' : 2p : : P : P 7 , gives, 



2p. P^ 16 16 



= p+p' = 2 + J 



. 3-3137085 



+ ^8 4-8284271 

 And from these numerical expressions for the surfaces 

 of inscribed and circumscribed polygons of eight sides, 

 we may evidently, by repeating the operation, and sub- 

 stituting the values just obtained for p and P, deduce 

 the numerical expressions for the surfaces of the sixteen- 

 sided polygons, and so on to any extent : the results are 

 as in the following table, which was in part given at 



page 569. 



No. of lido. 



4 



8 



16 



32 



64 



128 



256 



512 



1024 



2048 



4096 



8192 



16384 



32768 



Surf, of circ. pot 



4-0000000 

 3-3137085 

 3-182.W7!) 

 3-1517249 

 3-1441184 

 3-1422236 

 3-1417501 

 3-1416321 

 3-1416025 

 3-1415951 

 3-1415933 

 3-1415928 

 3-1415927 

 3-1415926 



Barf, of ins. pol. 



2 OOOOOOO 



2-8284271 



3-0614674 



3-1214451 



3-1365485 



3-1403311 



3-1412772 



3-1415138 



3-1415729 



3-1415877 



3-1415914 



3-1415923 



3-1415925 



3-1415926 



From these numerical values for the surfaces of the 

 inscribed and circumscribed polygons, it appears that 

 when the number of the sides is so great as 32768, the 

 two polygons differ so little from one another, that their 

 numerical measures, as far as seven places of decimals, 

 are absolutely the same. Now the circle, with which 

 these polygons are connected, is manifestly between the 

 two, as to amount of surface ; being greater than the 

 inscribed polygon, and less than the circumscribed one ; 

 consequently the surface of the circle must differ less 

 from that of either polygon than the polygons differ 

 from each other ; and as the polv^ons themselves differ 

 in numerical measure only, after the seventh decimal of 

 the number 31415926, <fcc. , it follows that this number, 

 as far as the decimals extend, is the numerical expression 

 for the surface of a circle whose radius is 1. By carry- 

 ing on the foregoing process, the expression for the sur- 

 face is found to be 3-141592653589793, &C. 



