SIGN. 



SIGH. 



dhtfawuiahed baa arithmetic; and it U difficult to pUoe it on any 

 tthfaotary bad* except tlut of distinct di-tiuitiuns not wholly d. r , . ,-"l 

 from arithmetic. On thin point, however, it is not our present purpose 

 to enter further ; the object of this article being the application ..f the 

 go*, and in particular thoai detail* of interpretation which an neces- 

 awy in the application of ordinary algebra- to geometry. By ordinary 

 algebra we mean that system in which the positive and negative 

 quantities are fully capable of interpretation, but in which */ 1 is 

 ooneidered aa incapable of interpretation. 



The relative meaning of + and U direct opposition of properties ; 

 and it U only where direct opposition b possible that complete inter- 

 pretation can exist. The symbol + 7 means not only 7 unite of its 

 kind, but 7 unite directed to be considered in a specific one of two (the 

 only possible) lights, or used in a specific one of two (the only possible* 

 manners; the first generally implying the second. Thus lot 7 inches 

 be measured from a given )>oint; the superposition of + or tells 

 nothing, for the measurement may be made in an infinite number of 

 different directions. Choose one of these directions, or rather one line 

 of direction, and the indeterminate character of the proposal (to 

 measure seven inches from the given point) is almost gone : there are 

 but two directions in which to do it ; if one of them (no matter which) 

 be signified by + 7, the other (no matter which, except with reference 

 to the first) must be denoted by 7. 



A problem may present different sets of oppositions of very different 

 kinds. Thus we might have a problem in which there are concerned 

 together 1 , time before or time after a certain epoch ; 2, height above 

 or height below a certain level ; S, the debtor or creditor side of certain 

 books. To give a more precise idea (it is hardly worth while to frame 

 a specific problem), a man might engage to build a wall on different 

 terms as to the foundation and what is above the ground ; for which 

 he might have to borrow money and pay interest up to a certain time, 

 when by receiving the whole amount due to him he might repay and 

 invest besides ; and the whole transaction might have to be pro|n -ily 

 entered in his books. The young student might suppose that if + 1 

 and 1 represent a foot of the wall above and a foot below the ground, 

 it will not necessarily follow that + 1 and 1 (undistinguishable from 

 the former) will do to represent one pound of interest due to and from 

 the contractor ; and still less that the same + 1 and 1 will also do 

 to direct It to be carried to the debtor or creditor side of his books. 

 But what he will learn from a properly established algebra (and until 

 he ha* learnt it, he is not in possession of any part of the difference 

 between algebra and arithmetic) is this that he would gain absolutely 

 nothing by inventing such distinctive symbols as would remove his 

 doubt of their applicability. Let ( + ) 1 and ( ) 1 represent feet of 

 wall, [ + ] 1 and [ ] 1 pounds sterling of interest, { + \ 1 and { } 1 

 pounds sterling carried to one side or the other of the books ; while 

 + 1 and 1 represent simple addition or subtraction. Let the problem 

 be fairly translated into algebraical language, and an equation formed 

 in which all the distinctive symbols are seen : algebra teaches that the 

 rules to be applied to that equation differ in no respect whatever from 

 those which would have been applicable if all the signs had come from 

 the same source of meaning. Perhaps it would be better if the student 

 were not allowed to come so easily by this result as he usually does, 

 but should be made to learn by his trouble how unnecessary the dis- 

 tinctions really are, aa to operations, and allowed in due time to feel 

 the relief afforded by dropping them. 



When a problem admits of but one opposition (say it simply refers 

 to time measured future or past from a given epoch), there is no diffi- 

 culty about the interpretation of any result If this result be positive, 

 it must be such time aa was called positive when the operation was put 

 into shape ; and the contrary. But in the application to geometry, an 

 extension of the interpretation of signs enables us to remove some 

 difficulties in the proper expression of angles, which we proceed to 

 describe. 



In the rectilinear figures considered by Euclid the sum of the 

 external angles is always equal to four right angles. In these figures 

 there are no re-entering angles. [SALIENT.] The proposition remains 

 true when there are re-entering angles, provided that the angles which 

 then take the place of what were called external angles be counted as 

 negative. In algebraic geometry it is usual to refer all points and lines 

 to two straight lines at right angles to one another. [ABSCISSA; 

 CO-OBDINATES.] The following conventions must now be employed : 



1. I.et v x and w Y be the axes of co-ordinates, meeting at o, the 

 origin. Let o x and o Y be positive directions : o v and o w, negative 

 directions. 



2. A line drawn from o to any point p is in itself (for the present) 

 neither positive nor negative ; either sign may be given to o r, but the 

 contrary one must be given to o <j. 



8. The line o p may, keeping its sign, revolve round o ; nor, if nega- 

 tive, U it to be counted positive when it comes into momentary coin- 

 cidence with o x ; nor, if positive, is it to be counted negative w -"in n in 

 coincidence with o v. And generally, a straight line revolving round 

 any point does not take the signs of lines which receive signs on 

 account of the fixed directions in which they are drawn. 



4. The angle made by two lines A and B is to have a distinction of 

 sign drawn, according as it is called the angle made by A with B, or 

 the angle made by B with A (the angles made by A from B, and by B 

 /row A, would be better). If the angle of A with B (say A A B) be pori- 



tire, then the angle mad* by B with A (B'A) i* negative, and MM 



r-- 



i 



5. The positive direction of revolution is that in which a line moves 

 from the positive part of the axis of x to the positive part of 1 1 

 of y (u marked by the arrows), 



! . Hie sign of any lino drawn through r is thus dct< ruiineil. If or 

 be potitire, that direction U positive in which the point r nm-t in>.\ > 

 so as to revolve jniatirrly ; thus, o v being positive, r K ia i 

 p L. negative. But if o r be negative, the reverse is the cane ; but the 

 rules need only be remembered which mippose o l' positive. 



7. When an angle amounts to more than four right angler, the four 

 right angles may be thrown away ; and generally, four right angles, 

 or any multiple of them, may bo added to or subtracted from any 

 angle. 



S. In measuring the angles made by two lines passing through 

 being positive), the positive directions on those linen (found a 

 must be used : and by A'B, the angular departure of A from i;. i- 

 umlerstood the amount of positive revolution which will bring n hit" 

 the position A. 



'.'. Hence it follows that A'B a either equal to A'X B*x, or differs 

 from it by four right angles, x standing for the axis of x. 



10. Hence also it follows that in every closed tiguiv, whether such 

 as those admitted by Euclid or not, some of the angles are nega; 

 every angle A A B be interpreted as A"X B"X. And in every such 

 measurement, the um of all the angles, with their proper signs, is 

 equal to nothing. But, measured as in (8), the sum of all the angles 

 is nothing, or a multiple of four right angles positive or negative. 

 This ambiguity is wholly indifferent in trigonometrical din-rations. 



To prove the last, let us consider a four-aided figure, of which the 

 sides are A, B, c, D. The angles of the figure, taken in order, a). 

 B"C, C"D, D'A, which, measured as in 9, are A"X B"X, n"x c.x, 

 D A x, r>*x A*X, the sum of which is obviously nothing. Hut if any one 

 of these angles should differ from the preceding, it can Ije onh 

 multiple of four right angles, whence the sum must be a multiple of 

 four right angles. 



We shall now take aa an example, the three angles of a ti i 

 estimating them first by each side with reference to the next, an. i 

 by roiii]iring each of the sides with the axis of x. Tin- M<; 

 marked A, B, o ; the angles of Kuclid, without reference to sign, are 



a, 0, y : and the positive and negative directions arc marked. Pour 

 right angles are denoted as usual by lit. Required A "B + B*c + C A A. 



In AlB, the amount of positive revolution by which the positive part 

 of B turns round into that of A is y + r. Similarly for B'c, we have 

 o + r, and for c*A, + r. The sum of these, a + + y being , is ir, a 

 multiple of 2ir. Now let the angle A'X be 6. Then B't.x will be seen 

 to be an angle in this figure greater than two right angles, and will ! 

 found to be Tt + 9 y, while c*x is greater than three right angl< 

 is 2r a-*- B y. Hence we have 



7) 9 



=)8-nr. 



Only the third angle gives precisely the same in both ; in the other 

 two, the second determination gives in each case four right angles less 

 than the first. The sum of the three angles is now 0. 



The use of thin system lies in enabling us to give in a general form 



