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RATIOS OF ANGLES UNLIMITED IN SIGN AND
§ 1. DIRECTED MAGNITUDES. 285. LET L and M be two points in a line: and let LM stand for the length of the part of the line between them.
Let T be any other point in the same line.
(1) If T is between L and M, LM = LT + TM ..(1), (2) If T is outside LM, beyond M, LM=LT - MT ...(2), (3) If T is outside LM, beyond L, LM = TM-TL ... (3).
If we do not know where T is situated with regard to L and M, we do not know which of these three formulæ to adopt. But it would be clearly advantageous to be able to use the same equation for all three cases. We observe, then, that
MT has to be subtracted, when Tis on the side of M, opposite to L: and that
TL has to be subtracted, when T is on the side of L, opposite to M: and that
Otherwise MT and TL have to be added.
286. Let us, then, use a bracketed symbol, such as (LM), to indicate—not merely the distance LM—but also the direction in which that distance is regarded as having been described.
Such a symbol as (LM) or (ML) may be said to represent a directed length. Thus
(LM) represents the line LM, regarded as drawn from L to M;
(ML) represents the line LM, regarded as drawn from M to L.
287. The use of these symbols representing directed lengths is explained by the following rules of interpretation.
To interpret any expression in which directed lengths in the same line are connected by the signs + or -,
Rule 1. When the lengths are all directed the same way, the signs must be understood in their Arithmetic sense.
Rule II. Any directed length may be reversed, if the sign preceding it is changed.
To obtain the final interpretation, therefore, we must reduce all the directed lengths to lengths directed the same way, by reversing and changing the preceding signs where necessary.
288. From the rules of the preceding article, it will be seen that we may write always in Art. 285 (LM) = (LT") + (TM).....
(4). For, in fig. 1, (LT), (TM), and (LM) are directed the same way, and hence + has to be interpreted as arithmetic addition.
In fig. 2, (TM) is directed in the opposite way to (LT) and (LM), and .: +(TM) becomes – (MT).
In fig. 3, (LT) is directed in the opposite way to (TM) and (LM), and .. + (LT) becomes – (TL).
289. The formula (LM)= (LT) + (TM) is of the highest importance. To give it a familiar aspect, we may regard (LM) as a step from L to M. Thus, if we have to go from L to M, we may always go from L to T and then from T to M.
This will be true, even if T is not in the line LM. But in this case LTM would form a (rectilineal or curvilinear) triangle, and we should not in general) be able to give any arithmetic meaning to the sign +, for neither the addition nor the subtraction of the lengths of two sides of a triangle would give us in general) the length of the third side.
If L, T, M are in the same line (straight or curved) we may always give an Arithmetic Interpretation to the equation (LM)=(LT) + (TM), by reversing and changing the sign of either (LT) or (TM), if it is in the direction opposite to (LM).
290. The student should observe that with the same rules of interpretation, the equations (LM) = (LT) – (MT)...
(5) and (LM) = (TM) - (TL)...
.(6) are always true, if L, T, M are in a line. In fact, by Rule 2,
+(TM)=-(MT) and + (LT) = - (TL), so that (5) and (6) follow from (4).
291. In the fundamental equation
+ (TM)=- (MT) write M for T. Then
+ (MM)=-(MM). Now the only algebraical quantity which is such that its addition is equivalent to its subtraction is zero. Hence
(MM)= 0. This, of course, only expresses the fact that the distance from M to M is nothing.
292. Similarly, if OL, OM make any angle LOM, and if OT be any other line through 0 in the plane LOM, the angle LOM will equal either
LOT + TOM or LOT – MOT or TOM – TOL, according as OT is between OL and OM, or beyond OM, or beyond OL.
Using the symbol (LOM) to represent the angle LOM, regarded as described by the revolution of a line from OL to OM, and giving the same rules of interpretation for directed angles as for directed lengths, we may say universally
(LOM) = (LOT)+(TOM). As before this equation is clearly realised by regarding the rotation from OL to OM as equivalent to a rotation from OL to OT followed by a rotation from OT to OM. But the equation can be arithmetically interpreted, only if OT is in the same (surface or) plane as the angle LOM.
293. If OLO', O'MO are two parts which together make up any closed line, and if OTO' be any other line joining 00', then the area of OLO'M will equal either
OLOT + OTOM or OLOT - OMOT or OTOM - OTO'L, according as OTO'falls entirely within the figure OLO'M, or
outside the figure beyond OMO', or outside the figure beyond OLO'.
Using the symbol (OLO'MO) to represent an area whose periphery is traced out by moving in the direction 0-L-0-M-O, and giving the same rules of interpretation for directed areas as for directed lengths and directed angles, we may write universally
(OLOMO) = (OLO'TO) + (OTOMO). In tracing out the peripheries of the two figures whose sum is equal to the original figure, the tracing point moves twice over the part of their peripheries which is common, viz. OTO';—first in the direction (OʻTO), then in the direction (OTO'). These two movements, being in opposite directions, may be regarded as cancelling one another, and the resultant movement as equivalent to (OLMO).
Algebraical Representation of Directed Magnitudes. 294. In each of the three kinds of directed magnitudes above discussed, -viz. length, angle, area,—two opposed directions were involved.
Now in dealing with such magnitudes, we have to reduce them to the same direction before combining them arithmetically.
Hence it is convenient to choose one of the two directions in question as that to which all the directed magnitudes shall be reduced.