522 



CURVE LINES. 



Curve if it does not lie all in one plane, it is called a curve of 

 Lilies. double curvature : the oblique rhumb lines on a com- 

 <W "V"*' mon terrestrial globe are curves of double curvature. 

 5. In the theory of plane curves, the first thing to 

 be considered is the manner of determining and descri- 

 bing the situation of a point on a plane. This may be 

 done in various ways. For example, we may estimate 

 its distance from two given points, and then its posi- 

 tion will be determined by the intersection of two cir- 

 cles described on these points as centres, with its known 

 distances from them as radii. Or we may consider how 

 far the point is from a given point, and also from a 

 straight line given by position, and either of these me- 

 thods will apply with advantage to the describing of the 

 conic sections ; for it has been shewn, that the ellipse 

 and hyperbola are lines of such a nature, that the sum 

 of the distances of any point in the former curve, and 

 their difference in the latter from two fixed points, is al- 

 ways a constant line ; also, that the distance of every 

 point in any of the three sections from a given point, 

 has to its distance from a given line a given ratio. These 

 methods, however, would not be found to be generally 

 convenient, and therefore their application is limited 

 to the particular curves referred to. 



G. There are, however, two other methods of general 

 and easy application. By the first, the situation of a 

 point is determined by its distances from two straight 

 lines, having given positions, and intersecting each 

 other, (commonly at right angles,) just as the position 

 of a point on the earth's surface is determined by its 

 latitude and longitude ; and by the second method, the 

 position of a point is determined by its distance from a 

 fixed point, and the angle which a line, drawn from 

 it to that point, makes with a line given by position 

 passing through the same point. We shall be most par- 

 ticular in explaining the first of these methods, or ra- 

 ther that method rendered somewhat more general, be- 

 cause the transition from it to the second is easy. 

 F&»TE 7- Let us suppose then, that .r AX and^AY, Fig. 2, 



COXX VI T . are two straight lines given by position, intersecting each 

 Vi £- 2 - other at any point A, and let M be any point in their plane. 

 Draw MP, MQ parallel to AX, AY, meeting them in 

 P and Q. The point M is manifestly determined, if 

 we know the lines AP and AQ, and their directions in 

 respect of the point A. The lines AX, AY produced 

 indefinitely, are called the axes. The line AP, the seg- 

 ment of one of the axes intercepted between a line 

 drawn from M parallel to the other axis, is called the 

 abscissa of the point M ; and the line PM or AQ is cal- 

 led its ordinate. The line AX is called the axis of the 

 abscissae ; and AY that of the ordinates. It is a matter 

 of indifference which of the two axes is taken for that of 

 the abscissae. Any abscissa, and its ordinate, are com- 

 monly called co-ordinates. The point A, from which 

 the co-ordinates are reckoned, is called their origin. It 

 is usual to denote any abscissa by x, and the correspond- 

 ing ordinate by y, and to call AX the axis of x, and AY 

 the axis of y. 



8. If the two indeterminate lines AP, PM, or x and 

 y, be supposed to have, for a certain position of the 

 point M, the values 



xz=.a, y=-0, 

 as by these equations the position of the point is deter- 

 mined, we may call them ike equations of the point. 



The abscissa AP being supposed to remain the same, if 

 the ordinate PM decrease, the point M will approach to 

 the axis AX, so that PM, or b, becoming at last = 0, M will 

 fall on P ; therefore the equations of any point P in the 

 axis of the abscissa are of this form, xz= a, y =0. If we 

 now suppose the ordinate PM to remain the same and 



Curve 



Lines. 



the abscissa AP to decrease, then M will approach to 



AY, so that at last QM vanishing, we have x=0, y—b, 



for the equation of any point in the axis of the ordi- _ » ~" 



nates. 



Lastly, If we suppose both AP and PM to decrease, 

 and vanish at the same time, then we have x — 0, y = 0, 

 for the equation of the origin of the co-ordinates. We 

 may therefore conclude, that by giving to x and y all- 

 possible values from to infinity, we can indicate the 

 position of every point whatever in the angle YAX. 



9. That we may see how the position of a point in 

 any of the other three angles made by the axes is to 

 be indicated, let us suppose, that instead of AY, ano- 

 ther line A'Y', Fig. 3, parallel to the former, is taken Plate 

 for the axis of the ordinates. Let A= AA', and let x' be CCXXVII. 

 the new abscissa, taken on the same axis AX, but reckon- F'g- 3 - 



ed from the new origin A'. If we now consider any point 

 M, situated in the angle YAX, we have AP=AA'-f 

 A'P,or x=zA-\-x'. But if we consider a point M' situ- 

 ated in the angle Y'A'A, and still represent its abscissa 

 A'P' by x', which denotes a variable quantity of any 

 magnitude whatever, we have AP'=rAA' — A'P', or 

 x=A — x' ; from which it appears, that if we wish to 

 render the same analytical formula x=A+x', applica- 

 ble at once to points situated in the angle XA'Y', and 

 to points in the angle A A'Y', we must for these last re- 

 gard the values of x' as negative, so that the change of 

 sign answers to their, change of position with respect to 

 the axis A'Y'. 



10. To confirm this result, and shew more clearly 

 how the preceding formula may connect the different 

 points of a plane, let us consider a point situated in the 

 axis A'Y' itself; then x' vanishes, and the formula 

 x=A-\-x' gives x — A ; and this is the value of the ab- 

 scissa AA' with respect to the axes AX, AY. But 

 if we wish that the same equation apply to points in 

 the axis AY, let us consider any one; point in that 

 line ; it is evident that its abscissa x is equal to 0, 

 and therefore the preceding formula gives A-j-.t'=:(), 

 or x'zz — A, Avhich is the value of the abscissa 1 AA' 

 supposing it referred to the axis A'Y'. The analytic 

 expression of this formula becomes therefore posi- 

 tive for the axis AY, and negative for the axis A'Y', 

 when the points of the plane are supposed connected by 

 the equation x—A-\-x'. This result applies equally to 

 negative values of x, and proves that they . belong to ' 

 points situated on the side of the axis AY, opposite to 

 that on which the positive values are taken : For M" 

 being supposed any such point, we may always draw a 

 new axis A"Y", which has the same relative position in 

 respect of AY as this last had in respect of the axis A'Y'. 



11. By removing the axis AX parallel to itself, and 

 fixing the new origin at A", (Fig. 4,) making AA"=B, f,>. 4, 

 and putting 1/ for the new ordinates, reckoned from the 



axis A"X", we shall have yz=B-\-y' for the points situ- 

 ated in the angle YA"X", and y=B — y' for those in the 

 angle AA"X" ; so that to comprehend both in the same 

 analytic formula, the negative values of y' must be con- 

 sidered as corresponding to points situated on the side 

 of the axis A"X", opposite to t that 'on which the posi- 

 tive values lie ; and as this applies equally to the axes 

 AX, AY, we may conclude, that the change of the sign 

 of the variable line y, answers to a change of position of 

 the points from one side of tiie axis of the abcissae to the 

 other. 



12. Upon the whole, it appears that the negative va- 

 lues of the co-ordinates must be taken in a direction the 

 opposite to that of their positive values, otherwise the 

 same formula cannot be applied to all the points of a 

 plane, but will only comprehend the points situated in 



