GEOMETRY. 



Y A X, then x = -\- a, y = -{ b 



The equation of M' is .v = — a, y =z -^ b 



M" is .1 = — ,1, y = — b 



M'" is J,- = + <7, ji = — i 



If the point M is fituated on the line A X, then ji = o, 



:r =; rt. 



If it is fituatcj on A Y, then .v = o ; j' = b, and at the 

 point A, .V = o ; _y — o. 



Tlie point M may likewife be defined by the length of 

 the line A M, and by the angle ?, which it makes with the axis 

 A X. If this length be cxprefled by z, and the angle by <f, 



z - c ; ?) = A 

 A° being th.e number of degrees coii'.ained in the angle f, and 

 e the value of -z. 



The pofition of a flraight line is determined in a plane by 

 the co-ordinates of two of its points, or by the co-ordinates 

 of a fingl ■ point, and the angle which it makes with one of 

 its axes : the latter is tlie metliod moll ufually employed. 



If tlie line pafTcs through the origin of the axes, its pofi- 

 tion is determined by the angle which it makes with the 

 axis. 



We fliall firll confider this cafe as being the mod fimple. 

 £.et it be propofed to determine the relation between the co- 

 ordinates of any point in fuch a llraight line. 



Let us take, for example, a point M [Jg 2.) whofe 

 abfcifla is A P' =:: x\ P'M'=^': thefe co-ordinates, hke 

 thofe of the other points, making an angle S with each other, 

 which is that of the axes, we Ihall have this equation ; 

 y' fin. a. 



and for the points M', M'", S:c. whofe co-ordinates would 

 be- x",y"; x'",y''', &c. we have 



fin. * y"' fin 



X 



y 



x" 



&c. 



fin. (o — a) x'" fin. (o — k) 

 If .V, ji reprefent the co-ordinates of any point in A L, the 

 general equation will be 



I l"-_fL__. (,) 



X fin.' (o — 7.) 



thus .1- becoming fucceffively x', x'. Sec. y changes to the 

 correfpcnding ordinates y', y", &c. The formula ( I ) is 

 called the equation of a ilraight line ; it is the algebraic 

 enunciation of a property common to all its points, and is 

 thus written ; 



fin. a. . . 



y=^(ryi)-' (^-) 



The abfcifTa .v being given, the ordinate y may be deduced. 



When the angle ■« varies, the line changes its pofition, 

 turning roimd the point A; thus another ordinate _)■ cor- 

 refponds to the fanie abfcifTa .v, which alfo happens if b varies, 

 but it is effcntial to obferve that for all tlie values both of a 

 andi?, the equation (2) retains the fame form. 



If it fhould be required to find the relation between the 

 co-ordinates of the point of a ftraight line R' L', fituated in 

 any manner whatever, then if A L be drawn parallel to 

 R' I^', paffing through the point A, we (hall obferve that 

 for the fame abftiflTr, .i' .every ordinate of A L, for example ; 

 P' M v.ill be augmented by the fame quantity A R ^= i ; 

 let^' -f b be reprefented by j, equation of R' L.' will be ac- 

 cording to ( 2 ) ; 



fm. a 



y = f 77, r X + b ...^ (3) 



fm. (t — cj 



Here a and b gife the pofition of the llraight line R'L' ; 



whOTi thefe quantities are known it may be conftrufted, bu6 

 if the line is fubjedlto any particular conditions, as, to pafs 

 through a given point ; to be either parallel or perpendicular 

 to a given line ; to pafs through two points, &c. y. and b 

 become unknown quantities, as we fhall fee immediately. 

 Let us fuppofe the quantities a, o, and b given, and that 



it is required to conftruft the ftraiffht line v = -^ — -j -r 



' ° lin. (^ — a) 



X + b •, the problem is reduced to finding two of thefe 



points ; we endeavour to find the points in which the liae 



cuts the two axes AX, AY, ifg. 3.) the interfeclion R 



being the only point of the line in which x =^ O, and R' 



the only point of the fame line in wliich y z= o ; we fup- 



pofe, fuccedivclv; v = o, _y — O, and v/e (hall find y ■= b ■= 



fin. (o -.V.) 



AR, 



fin. 



^ = A R' J taking, then, A R' 



on the other fide of the point A, if it is negative, and A R 

 on A Y if it is pofitive, the line drawn through thefe two 

 points will be that belonging to the equation. 



If the equation to a flraight line ji = j.' -f- I be fuppofed 

 to refer to two axes, making an angle of 45 ' with each other, 

 and the inclination, a, which the hne makes with the axis X 



be required ; fince _ = i> fm. a = — rzr cof. a 



45 — « \' 2 



fin. 



fin, a ; confequently, tang, a r= 



V 2 1 + a' 2 



The value of b remaining conftant, the line takes every. 

 pofTible pofition round the point R ( /^. 2.) for every 

 pofiible angle from o to 360 ; for every angular value of 

 K, taken with every ordinate b, pofitive and negative, the. 

 line will pnfs through every point of the axis Y Y'. There 

 exills, therefiire, no line in the fame plane that cannot be 

 defined by equation (3), provided b and a are taken of a pro- 

 per value. 



The angle /S has no influence on the pofition of the line ;' 

 the variations of this angle onlv affeft the inclination of the 

 ordinate upon the axis of the abfcifTa? ; fo that for the fame 

 abfcifTe, the ordinate corrcfponds with another point in the 

 line. 



But, in general, the co-ordinates are fujipofed rectangular j 

 in which eafe, fm. p :=z fin. 00" =- I, and fin. (S — a^ = 

 cof. a ; and equation (3) becomes y =; x tang, .x -|- i =r 

 ax -'t-b, a being fuppofed the t.mg. . . 



When the flraight line pafTes tln-ough the origin of the 

 co-ordinates, its equation becom s j' ^^ ax. 



This equation is conflruclcd by taking A P = i = 

 radius, and then fetting oiT from a fcale of equal parts the 

 value of a on tiie perpendicular P M {^f.g. \.'^ ; PM being 

 equal to a, M will be a point in the required line. 



We may now conftruft the equation ji = .i- -f \, y ■=: 



— X — I. 



Thefe two lines cut the axis A X in the fame point, and 

 are fituated fimilarly to it, one above, the other below ; they 

 are moreover perpendicular to each other. 



The flraight lines y ^=. — .v — i,_y =: — -r -f 1, are pa- 

 rallel, bccaule they make the fame angle with the axis,, 

 having the fame tangent, — i. 



The flraight line exprefTed by the equation _y =r .»• a^ — i. 



— 1 is reduced to a point on the axis Y below the 

 origin, and diflant from it by a quantity equal to unity, 

 fince, for ever)' other vnlue than zero, the ordinate is imagi- 

 nary. 



Tlie 



