ISOPERIMETRY 



iYzw C B ; ati(3 alfo from any otVier points C, C", afTumcd 

 in L M, draw C A, C 15,, C D ; then it is obvious, that 

 CD, CD,-C"D, are rLfpeftively equal to C B, C B, 

 C" B, and therefore A C + C B = A C -f CD, and 

 AC4-CB-AC + CD = AD; and confcquently, 

 C\.>-e two fides of a triangle are pfi-eater than the third 

 f.de, we ha^^e AC + C' D > AD, or (A C 4- C B) 

 > (AC + C B) i aad the fame is true of any other point 

 C in the line L M; therefore A C + C B is lefs than 

 anv other two lines that can be drawn from _A, B, to 

 meet i:i the line L M, and confeqnently A B + AC + 

 C B is the triangle, having the minimum perimeter, and it 

 lias its fides meeting L M at equal angles, as is evident, 

 Q . E . D. 



Cor. — Of all lines draw'n from two gi\Tn points, to meet 

 in a line given in pofitioii, the fum of thofe two fliall be 

 the leafl, that make equal angles wirh the given line. 



Prop. HI. 



Of all triangles having the fame bafc, and the fum cf the 

 o her two fides tlie fame, the. ifofceles is the greatelL 

 /■,-. 3. •■ , : 



Let A C B be an ifofceles triangle on tlie Tiafci A.B, 

 and ADB a triangle on the fame bafe, havinij its two 

 fides A D ^- D B =- A C + C B ; then will A B C be 

 tlie crr-'atell trian-^le. 



FTrll draw C H perpendicular to A.B, and DEF p.i- 

 rallel to A B, interfering 'C H (produced if neceffary) 

 in The point E ; likewife let A E and B^E be drawn. 



Now it is evident, that the angles A E F and B ED are 

 •equal, and confeqnently by Prop. II. A E + E B is lefs 

 than A D 4- D B, o'r lefs than the equal fum A C + 

 C B ; therefore the point E, and confequentl.y the whole 

 triangle A E B, mull fall within the triangle A C B., and 

 therefore the trianirle A E B, or its equal ADB, is lefs 

 thanACB. Q.E.D. 



Prop. IV. 



Of all triangles (landing on the fame bafe, and having 

 the fame vertical angle, the ifofceles one is the greatell. 

 Fig. 4. 



For fiiice all triangles whofe bafes and vertical angles 

 are equal have their vertices in' the fame circular fegment, 

 it is obvious, that the ifofceles triangle A B C is that 

 which has the greatelt perpendicular; and fince triangles 

 whofe bafes are given, are as their perpendiculars, it fol- 

 lows that the ifofceles triangle, which has the greated per- 

 pendicular, vvnil alio have the greatcit furface. Q. E. D. 



Phoj-. V. 



Of all right lines that can be drawn through a given 

 point, between two right lines given in pofition, but not 

 parallel, tliat which is bifeftcd by the given point forms, 

 with the other two lines, the leaft triangle. Fig. ^.^ 



Let A B, B C be any two lines given in pofition, and 

 D the given point ; then J fay that the line E D F, which 

 is bifecled in the point D, makes with the two given liies 

 A B, B C the kail triangle. For if E I be drawn parallel 

 to B C, meeting G H in 1, the equi-augnlar triangles 

 D F'H, and DEI, will be ecjual, becaufe ED = FD-; 

 and D F H will therefore be greater, or lefs, than DEC, 

 according as B G is lefs, or greater, than BE; in the 

 b'.ter cafe, let the fpace D E B H be added to both, fo 

 ih^ll F E B be lefs than G H B ; and if in the former cafe, 

 D G B F be added, then will H G B be greater than 

 F E B ; and confcquently FE3, in .tllis cafg .^Ifp, Ipls 

 DisnHGB. Q.E.D, 



Cor — If DM and D N be drawn parallel to B C and 

 B A, the two equal triangles D E M and D F N, taken 

 together (fmce EM = D N = M B), will be equal to thi^ 

 parallelogram D M B N ; and therefore this parallelogram 

 is equal to half the triangle FEB, but lefs than half tlie 

 trian'^le B G H ; whence it follows, that a parallelogram is 

 alivays lefs th.tn half the triangle in which it is infcribed, 

 except when the bafe of the one is half the bafe of the 

 other, in wtiich cafe the parallelogram is exafily equal to 

 half the triangle, which is the maximum parallelogram that 

 can be infcribed in any triangle. 



Ssf}ifi:'m. —Yrom the preceding corollary it might be de- 

 momlviited, that the leaft triangle that can pofiibly be de- 

 fcribed abont, and the greatelt parallelogram that can be 

 infcribed within, any curve, concave to its axis, will be when 

 the fubtangcnt is equal to Tialf the bafe of the triangle, or 

 to the vAolc bafe q£ the parallelogram. 



Prop. VI. 



Of all right-lined ligu-es, cont-ained under the fame 

 number of fidos, and infcribed in the fame .circle, that is-- 

 the greatell vvhufe fides are all equal. Fig. 6. 



F«r, if paflibls, let fome polygon, AB'C.EF., whofe 

 fides C E, F E, are unequal, be tlie greatell ; and let 

 'CDF be an ifofceles triangle, defcribed in the fame feg- 

 ment with CE F whicli, being greater than CEF by 

 Prop. IV.. the -whole polygon ABCDF is greater than 

 the polygon A B C E F, whereas we have fuppofed the 

 latter tu be the grcateft, which is abfard ; therefore tlve 

 polygon which has all i'.s fides equal is the greateil. Q. 



E. b. 



Cor. I . — It follows, with reference to the fame figure, 

 that of all right-lined figures, contained under the fame 

 perimeter, and of the fame number of fides, the greateil is 

 thitt which has all its fides equal. 



For if A B C E F be fuppofed the greatell, in wbick 

 the fides C E and E F are unequal, then the triangle CEF 

 would be greater than the triangle CDF, the fum of the 

 fides in bolh cafes being equal; but, fince CDF is an 

 ifofceles triangle, it is greater than any other on the fame 

 bafe, and of equal perimeier. Prop. III. ; therefore D C 

 and D F mufl be equal, and the fame may be demonllrated 

 of any other two unequal fides. 



Cor. 2. — Hence again it follows, that of all right-lined 

 figures, contained under the fame number of fides, and of 

 equal perimeters, the gi-eatett is that which may be infcribed 

 in a circle : the figure being a regular pclygon by the fore- 

 going corollar)-. 



Pjjop. VH. 



Of all reftilinear figures, in w^hich all the fitles except one 

 are given, the greateil is that whidh may be infcribed in a 

 femicircle, whofe diameter is that uiiknoun fide. Fig. y. 

 ■ For conceive ABCDEFtobca rrtftilincar figure^ that 

 is not iiifcribablc in a femicircle, and draw any two lines 

 A D, F D, from the extremities of the fide A F, to any 

 angle of the figure ; then it is obvious, th.it the whole figure 

 will be tile grcateft when the triangle A D Fis the greattft, 

 but this will be Prop. I. when A D F is a right angle, antl 

 in the fame manner we fmdtbat A C Fis a right angle, and 

 fo on ; aud confcquently when the figure is iirfcribed in a fe- 

 micircle, of whic-h the unknowm fide is the diameter, its fui> 

 face wiU be the grcateft. Q. E. D. 



Pnop. VIII. 



Of all figure?, made with fides given in number and mag- 

 ^ nitude, 



