GEOMETRY. 



and the diftancc fi-om this point to the horizontal plane 

 Y A X is M M"' or M' OT or A w'. Thus thefe diftanccs 

 may be found on the fixed hnes AY, AX, A Z. 



The points M', M", M'", of the perpendiculars let fall 

 from the point M on the planes to which we refer tlie pofi- 

 tion of this point, are called the vertical and horit^ontal pro- 

 jeaions of the point M, vertical in confidering M' and M", 

 and horizontal when confidering M'". 



Two of thefe projeftions are fufficient to determine the 

 point ; for if from each of them perpendiculars be drawn to 

 the plane which contains ft, they will interfeift in the point 



M. , ^ , 



The third projeftion evidently refults from each of the 

 two others, as may be feen in tlie figure. 



Since the poiition of a point is completely defined by its 

 diftance from three redangular planes of projeftion, if we 

 denote the diftance of the point from the plane Y A Z by .v, 

 the diftance of the fame point from the plane Z A X by y, 

 and its diftance from the plane Y A X by s, ai;d the linear 

 values of thefe diftances be denoted by a, b, c, its pofition 

 will be indicated hj x — a, y = b, -^ — c : the formula 

 X = a, z = c, denote the pofition of the pro]e6tion M : 

 the formulx y = b, z = c, denote the projeaicn M" ; 

 and thefe two projeftions are fufficient, as we have feen, to 

 find the pofition of the point ; and thefe two fyftems of 

 formula; comprife the data of the three diftances. When 

 the point is in the horizontal plane, z = O, and its pofition 

 IS fixed by .v = a, y =^ b : when it is in the plane Z A X, 

 V = o, and its pofition is determined by .v = a, % = c ; 

 and if in the plane Z A Y we have .v = o, and it is defined 

 by V = i, % — c. 



For a point fituated on the axis A X, we have a = O, 

 y = o, X = a. 



If on the axis AY, z = o, x = o, y = b ; and if on 

 AZ, X = o, y = o, z = c. 



At A, the origin of thefe diftances, we have x =: o, 

 y =. O, z = O. 



Every point in the plane, M M' mM'", and confequently 

 the plane itfelf, is denoted by .v = a, becaufe a is tlie com- 

 mon diftance of each of its points from the plane Y A Z. 



The plane M M" m" M" is denoted by y ~ b, and the 

 plane M M' m' M ' is Jefcribed by z = c. 



The pofition of thefe three planes gives that of the point M, 

 and confequently this point will be defined, as we have 

 already faid, by the formulae x = a, y — b, z = c. 



Of the Equations to a Jlraight Line. — The equations of a 

 ftraight line, fituated in fpace, exprcfs the relation which 

 cxifts between the co-ordinates .v, y, s, of any point what- 

 ever of the ftraight line : let us fuppofe it projected on the 

 planes xz, and yz ; thefe projections will be other ftiaight 

 lines, which have for their equations 



X — az -\- 'J. . . . . y = bz -\- P : 

 dimiaating z from thefe equations, the refulting- equation is 

 I) X a_y = ba — 01.15, which belongs to the plane xy. 



The equations of thefe three projeftions, of which any 



two imply the third, are the equations of tlie ftraight line, 



whofe pofition in fpace depends on -the conftant quantities 



ab, x0. 



To obtain the eo-ordinates of the points in which this 



■ftraight line cuts the three planes, we nuift make fuccefllvely 



the three values x = o, y — O, z — O, which gives x = «, 



y=§, for the point where the ftraight line interfects the plane 



B « /3 , , . . 



X ■= r- + « for ttie point where it 



interfefts * a ; « = — — , y = — -|- ;3 for the point 

 a a 



where it meets the plane y z. 



The ftraight line, whofe equation is x ^ az + a, makes, 

 with the axis z, an angle, whofe tangent ha: it cuts tlie 

 axis X in a point, whofe diftance from the origin of the co-or- 

 dinates is equal y., fince, if in this equation s := o, a' = a. 



If two ftraight lines are fituated in the fame plane ; fup- 

 pofe that of X, z, then let the equation to the firft be 

 X = az -\- X, and to the fecond x = a' z + a' ; for thefe 

 ftraight lines to be parallel, a' muft := a, and, if perpen- 

 dicular, I -)- a «' = o, or a' = . 



a 



The equation of two ftraight lines, fituated in fpace 

 beiag, for the firft, 



X =^ a z + a, y z= b z + 13 

 X = a z + lyJ, y = b' z -f /3' ; 

 the equation, which exprefTcs that thefe lines meet each other, 

 is ('i' — a) (i — b) — ((S' — iS) {a' — a) =z o, which re- 

 fults from the elimination of .r, y, z, from the four equations 

 of the two ftraight lines. 



Problems relating to a Jlraight Line. — Prob. I. To draw 

 a ftraight line through a point given in fpace parallel to a 

 given ftraight line. Let the three reftangular co-ordinates 

 be .V, y, z ; z being iuppofed vertical ; and let the equa- 

 tions of the projeftions of the ftraight line on the vertical 

 planes he .v ~ a z + b, y ^ a' z -j- b ; then the equation 

 of the horizontal projeftion will he ay — a .v = a i' — a' b'. 



If k', y', z, reprefent the co-ordinates of the given point, 

 the equations of the required hne vrill be 

 X — X .= a (3 — z') 

 y — y' — a' (z — a') 

 a! {x — -v) = a{y — y) 

 of which any two imply the third. 

 Prob. II. 

 To determine the equation of a ftraight line draws 

 through two points given in fpace. Let x\ y', a', be the co 

 ordinates of the firft point ; x", y", z", thofe of the fecond, 

 the ftraight line pafling through the firft point, its equations 

 will be of the form 



X — .v' =z a (z — e') 

 y - y' = b (z - z'} 

 (See Prob. I. of the plane.) 



And fince it muil pafs through the fecond, its equations 

 muft alfo be 



X — .v'' := a (z — 2") 



a and i being eliminated from their four equations, the equa- 

 tions of the required ftraight hne v^ill be 



.r (z' - z") = z («' — .r") + x"z' - x' z" 



y{z' -z") = z{y'-y") +y'z'-y'z". 



The cq-ordinates of the two extremities of a right line 



being x', y', z", for the firft, and x'',y", z", for the fecond, 



the diftance between the exremities, or length of the line 



joining them, will be 



V'(x' - x'Y + (y - y")' {z' - z") . 

 Pkob. III. 

 To determine the conditions requifite for two ftraight 

 lines to meet in fpace. Let the equations of one ftraight 

 line be 



.\- = a 2 -f- a 

 y =^ bz + S 



and of the other 



*•.)' 



i 



X 



a z -\- a.' 

 b'z +,0' 



