TRANSACTIONS OF THE SECTIONS. 



Illustration of the Mean'mg of the doubtful Algebraic Sign in certain 

 FormultE of Algebraic Geometry. By Professor Stevelly. 



The author had been led some years since to see the importance 

 of the present question as bearing upon the determination of geo- 

 metric positions by algebraic symbols, by finding that, when trans- 

 forming the axes of coordinates, it was sometimes requisite to use 

 the positive sign for the perpendicular let fall from a given point 

 upon a given line, and at other times the negative sign, although no 

 intelligible reason for the difference was assigned in the books, nor 

 could he for a long time give any reason that was satisfactory to his 

 own mind, or which would lead to an unvarying rule. At length, 

 while reflecting upon the origin of this doubtful sign, he was led to 

 a conclusion which was quite satisfactory to himself, and which fur- 

 nished, he conceived, a complete key to the interpretation of this 

 and many similar cases. 



A' P 



It is well known that if A A^ and B B' be the axes of coordi- 

 nates, O being the origin, and it be arbitrarily determined to con- 

 sider distances measured from O towards B^ as i)ositive, it is nfe- 

 cessary, by the connection between algebraic addition and subtrac- 

 tion, and the increasing and diminishing of such distances, to distin- 

 guish by the negative sign all distances measured from O towards B. 

 A similar rule holds for the axis A A\ and for every other axis 

 passing through O ; from hence it can be readily shown that all 

 lines drawn parallel to any fixed line, such as A A\ and falling 

 upon and terminated by B B\ must be similarly distinguished ; 

 those that fall upon the upper side or face, for example, being sup- 

 posed to be positive, these falling upon the under side or face, must 

 be marked as negative. A similar rule can be easily shown to hold 

 for any line in the plane of these axes. And this being attended to 

 will inform us why algebra ought to mark with the sign + , the per- 

 pendicular let fall from a point (p) whose coordinates are (a?^ y') 

 upon a line whose equation is y=aa;+ b. Although at first we 

 should think that as but one point can have these coordinates, and 

 one line only have that equation, there can be but one perpendicular 

 to whose value we ought to be led ; yet, in fact, we find the per- 

 pendicular to be + y —(tx ,j,j^^ reason why algebra leads 

 — V 1 + a- 



