ISOPERIWETRY. 



ailtude, tliat which may be iufcribeJ in a circle is the greatcil. 

 JVkj. 8. and 9. 



Let ABCDE, &€., and abcde, &c. be two poly- 

 gons, of which the fides of the one are refpedively cqiial to 

 the fides of the other ; that is, A B = « i, B C = i f , 

 CD = cd, Si.c., thetirll of which is infcribed in a circle, 

 but the other not infcribable ; then I fay, that the polyjjon 

 ABCDE, &c. that is infcribed in a circle, is greater tiun 

 the polygon a Ic de, &c. which is not fo. 



For draw the diameter E P ; join A P, B P ; and upon 

 ■ab = A'Si make the triangle ap b equal in all refpefts to 

 the triangle A P B, and join ep. Then of the two figures 

 ed c bp,s.ndpagfe, one at lealt is not by (hyp.) infcribable 

 in a femicircle, of which ep is the diameter, confeqiiently 

 one at leaft of thcfe tv^o figures is fraaller tlian the corre- 

 fponding part of the figure ABCDE, Sec. ; and therefore 

 the whole of this lail figure ABCDEFGPis greater 

 than the other vvliole figure abc d efgp ; and if from each 

 of thefe there be taken away the equal triangles A P B, 

 and apb, there will reipain the polygon ABCDE, &:c. 

 greater than the polygon a be de, S^c. Q. E. D. 



Cor. — The magnitude of the greatell polygon which can 

 be contained under any number of unequal fides, does not 

 at all depend upon the order in wliich thofe lines are con- 

 nefted with each other. For Gnce in all cafes it muil be in- 

 fcribable in a circle, it may always be divided into the fame 

 number of ifofceles triangles, which will be refpeftivcly 

 equal in all cafes. 



Prop. IX. 



Of all pol)-gons clrcumfcribed about the fame or equal 

 circles, that has the greateft lurface which has the gieateft 

 perim.eter. For conceive radii to be drawn from the centre of 

 the circle to each of the points of contact, tlven it is obvi- 

 ous that the area of the polygon will be equal to the reClan- 

 gle of the radius into lialf the perimeter of the figure; there- 

 fore the area being as the perimeter, it follows that the poly- 

 gon, having the greateit perimeter, will have the greateft 

 area. Q. E. D. 



Cor. 1. — Hence the area of any polygon circumfcribed 

 about a circle, is to the area of the circle, as the perimeter 

 of the former is to the circumference of the latter. Alfo, 

 the area of different polygons circumfcribed about the fame, 

 or equal circles, are to each other as their perimeters. 



Cor. 2. — If a circle and a polygon, circumfcribable about 

 another circle, are ifoperimeters, they will be to each other 

 as the. radii of the circles. 



Prop. X. 



The circle is greater than any reftilinear figure of the 

 fame perimeter ; and it has a perimeter lefs than any reilili- 

 near figure of equal furface. Figs. 10. and 1 1. 



Let the circle P Q, and the polygon A B C D E F, be 

 ifoperimeters ; then 1 fay that the circle is greater than the 

 polygon. 



For firft, whatever may be the number of fides of the poly- 

 gon, it will be the greatell under the fame perimeter when 

 the polygon is regular, Cor. 1. Prop. V I. ; and confequently 

 there may be a circle infcribed within it, the area of which 

 circle will be evidently lefs than the area of the polygon, and 

 therefore by Cor. I . of the preceding propufition, the circum- 

 ference of this circle will be lefs than the perimeter of the 

 polygon, or lefs than that of the circle P Q ; and confe- 

 quently, alfo, the radius of the former will be lefs than the 

 radius of the latter. But by Cor. 2. Prop. IX. the area of 

 the circle P Q, is to the area of the polygon, as the radius of 

 the circle P Q is to th* radius of the circle a b ; and there- 



fore the area of the formw is greater tlian the area of tin? 

 latter ; that is, a circle is greater than any right-fined figure 

 of ecjual perimeter. 



Again, converfely, if the areas are equal, the circumfer- 

 ence of the circles is lefs than the perimeters of the poly- 

 gon. 



For conceive a circle to be made whofe c-ircumferciice is 

 equal to the perimeter of the polygon, then will tins, circle 

 be greater than the polygon by v.liat is prov'.d above, ami 

 coniequently greater than that circle which is equal to thff 

 poly.gon; and therefore its circumference will alfo Vk- greater, 

 that is, tlie perimeter of the polygon \\\\\ aKvays be greattr 

 than the ciicumfcrcnce of a circle of equal iurface. 

 Q.E.D. 



Pitop. XL 



The greateft reftangle, that can be contained under the 

 two parts of a line, any how divided, will be when the lint; 

 is bifeaed. 



Let A B be a line that is bifefted in C, th»n will .\ C .-: 

 CB be greater th.an A D x D B, D beii-g any other 

 point in the line A B. 



For A C X C B = A CS but A D x D B =: [\ C 

 — DC) X ( A C + D C) = A C - D C ; confequeut- 

 ly the firft redtanglc is the grfateft. Q.E.D. 



Pkop. XIL 



The greateft folid that can be contained, under the three 

 parts of a given line, any way taken, will be that in whitli 

 the three parts are equal to each other. 



For fuppofing the point C fixed: then the reflangle of the 

 two parts AD X DC, will be the greateft when A D = 

 D C (by Prop. XI ). In the fame way, if any other poi;it 

 be fuppofed fi.Ked, as D, then will the reftangle D C x C B 

 be the greateft when D C = C B, and confequently the 

 folid ADxDCxCB will be the greateft when thefe 

 parts are all equal. O. E. D. 



Cor Hence of all parallelopipednns having the fum of 



their dimenfiims the fame, the cube is that which has the 

 greateft capacity. 



Phop. XIII, 



A line being divided into two parts, the folid that is con- 

 tained under one of thofe parts, and the fquare of the other, 

 will be the greateft when the latter part is double the 

 former. 



Let A B be divided into two parts in the point C, mak- 

 ing A C = 2 C B, then will AC" X C B be greater than 

 when C is taken in any other part of the line .A B. 



For >n whatever part of the line A B the point C be 

 taken, the point A C may be bifefted in D, and then we (hall 

 have AC- x C B = four times AD x D C x C B, but 

 thislartisthegreateft whenAD= DC = C B(Prop. XII.) 

 therefore the former is the greateft when A C = 2 C B. 

 Q. E. D. 



Piiop. XIV. 



Of all prifms of equal altitudes, and whofe bafes are alfo 

 eqnal, and like, the right prifm lias the fmalleft furface. 



For the area of each face of the prifm is propor- 

 tional to its height ; and therefore the area of each face is 

 the finalleft when its height is the fmalWfl, that is, when it 

 is equ.il to the altitude of the prifm, which is evidently wiieii 

 the prifm is a right one. Q. E. D. 



Cor. — And hence, converfely, of all prifms wliofe bafcs 

 are equal and like, and whofe lateral fiurface is the fame, 

 the riglit prifm has the great.ib altitude and capacity. 



3 O 2 FiiOP. 



