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45. IN a science which deals with the distances measured from a fixed point it is convenient to have some means of distinguishing a distance, measured in one direction from the point, from a distance, measured in a direction exactly opposite to the former. This contrariety of position we can denote by prefixing the Algebraic signs + and -- to the symbols denoting the lengths of the measured lines. The following illustration may serve as an introduction to this important theory.

46. Suppose that two straight roads NS, WE intersect one another at right angles at the point 0.

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A traveller comes along SO with the intention of going to E.

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Suppose OE to represent a distance of 4 miles,

and OW to represent a distance of 4 miles,

and suppose the traveller to walk at the rate of 4 miles an hour.

If on coming to O he makes a mistake and turns to the left instead of the right he will find himself at the end of an hour at W, 4 miles further from E than he was when he reached O.

So far from making progress towards his object, he has walked away from it: so far from gaining he has lost ground.

In Algebraic language we express the distinction between the distance he ought to have traversed and the distance he did traverse by saying that OE represents a positive quantity and OW a negative quantity.

47. Availing ourselves of the advantages afforded by the use of the signs + and - to indicate the directions of lines, we make the following conventions:

(1) Let O be a fixed point in any straight line AOB.

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Then, if distances measured from O in the direction OA be considered positive, distances measured from 0 in the direction OB will properly be considered negative.

The direction in which the positive distances are measured is quite indifferent, but when once it has been fixed, the negative distances must lie in the contrary direction.

(2) Let O be a fixed point in which two lines AB, CD cut one another at right angles.

If we regard lines measured along OA and OC as positive, we shall properly regard lines measured along OB and OD as negative.

This convention is extended to lines parallel to AB or CD in

the following manner:




Lines parallel to CD are positive when they lie above AB,

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Lines parallel to AB are positive when they lie on the right

of CD, negative when they lie on the left of CD.

48. We may now proceed to explain how the position of a

point may be determined.

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NS and WE are two lines cutting each other at right angles in the point Q.

The position of a point P is said to be known, when the lengths of the perpendiculars dropped from it on the lines NS and WE are known, provided that we know on which side of each of the lines NS and WE the point P lies.

If the perpendicular dropped from P to WE be above WE it is reckoned positive.

If the perpendicular dropped from P to WE be below WE it is reckoned negative.

If the perpendicular dropped from P to NS be on the right of NS it is reckoned positive.

If the perpendicular dropped from P to NS be on the left of NS it is reckoned negative.

49. about

Let a line QP starting from the position QE revolve in the direction ENWSE.




Then all angles so traced out are considered positive.

When QP reaches the line QN it will have traced out a right angle,

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If we suppose the line to revolve in the direction ESWNE, we may properly account the angles traced out by it to be negative angles.

For the sake of clearness we shall call QP the revolving line, and QE the primitive line.

50. Now suppose P to be a point in the revolving line QP.

Let a perpendicular let fall from P meet the line EW in the point M, and let this be done in each of the four quarters made by the intersection of NS and WE, as in the diagrams in the next article.

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Note. When we say that a line PM is positive or negative we mean that its measure has the + or - sign prefixed to indicate the direction in which it is drawn.

51. Let the line QP revolving from the position QE about Q describe the angle EQP, which we shall call the Angle of Refer


From P let fall the perpendicular PM on the line EQW.

We then obtain a right-angled triangle PQM, which we shall call The Triangle of Reference,

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