■GEOMETRY. 



tliirJ equal to zero^ we thiie find the equation of condi- of the oppofite fides, tJie trigonometrical tangent will be 



= —- — ;;, a'" = -r-^ r,i- Thefubftitutionofthcfe 



tion 



a'[y'-y'")+a"(y"-f) + a"{f-/) + a'a"(x'-x")=.o[l) 



+ a"fl"'(x'-.f"') 

 + a"'a'(x"'-x'} 

 Let us fiippofe the required ftraight linc^ lobe porpondi- 



cular to the fides of the triangle, and then examine if, upon 



tliis fuppofition, the equation ( I ) fubfills. 



Values in equation (3) will give, after dividing by y, 

 , f y _ x" - x' 7 



" l.v"'+ x' "^ x' - 2>J 



The equation to B C is y — y" — 



y - y 



and that of the ftraight line drawn tlu-ough D is _)> 



y'=a' 



.v'" -f- X ' 



; + 



y X" 



x' + 2x"' (x"' + x'){x'-2x"') 



; from whick 



we deduce a' = 



2/ 



; and as this is found to be the va- 



lue of a', tliat is, of the trigonometrical tangent of the anr 



Now as this latter ftraight line fliould be perpendicular to gie ^vliich tlie hue D M makes with A X, it may be con- 



the other, the equation aa' + l — o, from whence a' = , 



ought to exirt between the trigonometrical tangents ; but 



y'- y 



-, therefore a' = 



here a = 



jc'' — X"' y — y" 



The ftraight line drawn througl\ B being perpendicular to 

 D C, we conclude 



y -y = 4;^y!i (■-■ - ■*■■). y-y" = "" (^ - =<") 



and the ftraight line drawn through C being perpendicular 

 to D B, we have likewiie 

 v' — v" , 



Hence the values of a", a'", are 

 x'" - x' 



'=«'" (.v-.v'") 



eluded that this third line paftes tlu-ough the point of inter- 



feftion of the two firft. 



Let us confider the perpendiculars drawn from the centres 



of the fides of the triangle ADC {Jij^. 9.), the equation 



r' 

 of A D is V = -, -v 



X 



ofDC,;--y=-^— (x-y) 



of AC, y= o 

 The equations of the perpendiculars to thefe fides are. 



y-iy 

 y 



{x - 1 *') 



x' - x'" / ..-v' -I- ,v"'\ 



iy- Y-{^-——) 



^in 



and like wife 



a'a"{x'-x")z= 



a"d"{x"-x")=- 



y'-y y'-r 



W-y')yy"-y'")h'"-y') ^ ^ ' 



N 



fl"«'(.v 



.v') = - 



D 



N_ 



15" 



-(/'-/') 

 </"-/) 



..(2) 



J 



The equation of condition may be immediately formed by 

 determining .v from the two firft equations, and employing 

 this value in the third ; which ought to be fatisfied by this 

 fubftitution if there exifts a point of interfcction ; now x is 



x'" 

 found =; — ; tlierefore thefe three lines interfedl in a 

 2 



point. 



N and D heuig the numerator and denommator of tlie frac- Meliod of ddermmwg the Pofithn of a Point m Space.— 



tion winch multiplies^'- J-". By the fubftitution of ail Let A X, AY, A Z, be tlu-ee ftraight hues rtciproodly 



tkfe values the equation of condition 13 fatished, fcr it be- perpendicular to each other (fg. ic.) at the point A ; each 



*'5''^ss of them will be perpendicular to the two others, becaufe it 



N, 



• (x'<- .v"'+ .v"' -.f'+ .v'-.v'O + fs^J-'-jt'" -^f-y" +y'" 



IS perpendicular to two ftraight lines which interfeft at it* 

 D "' "* ' ■' "' ' "" extremity in this plane. Therefore each of thefe planes will 



_y' 1 = o ( /%•. 8 ). If the fide B C be placed on the axis of the ^•-' ="^ ''j^ fame time perpendicular to the two others. Thefe 

 Hbfcid'a;, which docs not alterthe general nature of the rcfults, ''""ee planes form then the three faces of a reftangular paral- 

 and alfo the point B be placed at A, then j." = o, x" := o, Wopipedon, and the fohd trihedral angle A. Let us fup- 

 y'"- o, and the equation of condition is fimplified, and be- P'-'''^ '"^^ planes Z A X and Z A Y vertical, and the p]an( 



a" y' -(- a'" y' + a' a" x' — a" a'" x"' + a'" a' (x'"—x') Y A X horizontal. Let a point in fpace be reprefented by 



= (3). In this pofition of the triangle, a' - '- M, fituated out of the planes 7. AY., 7. A Y, Y A X in 



,, _ .,„ real pofition, for example, before the firft plane, to the right 



p = 00, and in fa£l the line D R is perpendicular to of the fecond, and above the third; and let us fuppofe 



y' — y'" - ' _ perpendiculais M M', MM", MM", -Vom the point M to 



AX. If in equation (3) the terms which do not include fl' thefe three planes ; thefe perpendiculars will meafure the 

 be fuppreffcd, to cxprefs that the tangent is infinite, the {horteft diftances from the point to each of thefe planes, 

 equation (3) will be fimpl'.-.ied, and exprefied, thus, The planes drawn through the perpendicular M M' and MM", 



M M' and M M'", M M" and M M'" will enclofe the paral- 

 lelopipecon, and the point M v>-iil be the fummit of thcfolid 

 trihedral angle M oppofite to the angle A. 

 y i y ■ The diftanceM M' from the point M, to the plane Z A X 



li^es the preceding equation is fatisfied. ■ is in real length equal to M' m' or A m" ; the diftance M M" 



If from the points A and C lines be drawn to the middle from the fame point to the plane Z A Y is M" hi'' or A n:, 



aad 



«' a" x' + a' ' ii' (.r'" — .v") = o ; and dividing by a', a' x' ■(- 

 «■" {.■«'" — x') = o ; but in this pofition of the triangle, 



x" - x' ... x' 



■we have a" = 



a' 



'" = — — ; and by thefe va- 



y 



