SECTION III. GEOMETRIC INTERPRETATION OF ALGEBRAIC FORMS. 154°. We have a language of symbols by which to express and develop mathematical relations, namely, Algebra. The symbols of Algebra are quantitative and operative, and it is very desirable, while giving a geometric meaning to the symbol of quantity, to so modify the meanings of the symbols of operation as to apply algebraic forms in Geometry. This application shortens and generalizes the statements of geometric relations without interfering with their accuracy. Elementary Algebra being generalized Arithmetic, its quantitative symbols denote numbers and its operative symbols are so defined as to be consistent with the common properties of numbers. Thus, because 2+3=3+2 and 2.3=3.2, we say that a+b=b+a and ab=ba. This is called the commutative law. The first example is of the existence of the law in addition, and the second of its existence in multiplication. The commutative law in addition may be thus expressed :A sum is independent of the order of its addends ; and in multiplication-A product is independent of the order of its factors. Again, because 2(3+4)=2.3+2.4, we say that a(b+c)=ab + ac. This is called the distributive law and may be stated thus :— The product of multiplying a factor by the sum of several terms is equal to the sum of the products arising from multiplying the factor by each of the terms. These two are the only laws which need be here mentioned. And any science which is to employ the forms of Algebra must have that, whatever it may be, which is denoted by the algebraic symbol of quantity, subject to these laws. 155o. As already explained in 22° we denote a single linesegment, in the one-letter notation, by a single letter, as a, which is equivalent to the algebraic symbol of quantity; and hence, A single algebraic symbol of quantity is to be interpreted geometrically as a line-segment. It must of course be understood, in all cases, that in employing the two-letter notation for a segment (22°), as “AB,” the two letters standing for a single line-segment are equivalent to but a single algebraic symbol of quantity. The expression a+b denotes a segment equal in length to those denoted by a and b together. Similarly 2a=a+a, and na means a segment as long as n of the segments a placed together in line, n being any numerical quantity whatever. (28°) a-b, when a is longer than b, is the segment which is left when a segment equal to b is taken from a. Now it is manifest that, if a and b denote two segments, a+b=b+a, and hence that the commutative law for addition applies to these symbols when they denote magnitudes having length only, as well as when they denote numbers. 156°. Line in opposite Senses.--A quantitative symbol, a, is in Algebra always affected with one of two signs, + or which, while leaving the absolute value of the symbol unchanged, impart to it certain properties exactly opposite in character. This oppositeness of character finds its complete interpretation in Geometry in the opposite directions of every segment. Thus the segment in the margin may be considered as extending from A to B or from B to A. A With the two-letter notation the direction can be denoted by the order of the letters, and this is one of the advantages а B В' of this notation; but with the one-letter notation, if we denote the segment AB by +a, we must denote the segment BA by – a. But as there is no absolute reason why one direction rather than the other should be considered positive, we express the matter by saying that AB and BA, or +a and – a, denote the same segment taken in opposite senses. Hence the algebraic distinction of positive and negative as applied to a single symbol of quantity is to be interpreted geometrically by the oppositeness of direction of the segment denoted by the symbol. Usually the applications of this principle in Geometry are confined to those cases in which the segments compared as to sign are parts of one and the same line or are parallel. Ex. I. Let ABC be any A and let BD be the altitude from the vertex B. Now, suppose that the sides AB and 'BC undergo a gradual change, so that B may move along the line BB' until it comes into the position denoted by B'. Then the segment AD gradually diminishes as D approaches A ; disappears when D coincides with A, in which case B comes to be vertically over A and the A becomes right-angled at A; reappears as D passes to the left of A, until finally we may suppose that one stage of the change is represented by the AAB'C with its altitude B'D'. Then, if we call AD positive, we must call AD' negative, or we must consider AD and AD' as having opposite senses. Again, from the principle of continuity (104°) the foot of the altitude cannot pass from D on the right of A to D' on the left of A without passing through every intermediate point, and therefore passing through A. And thus the segment AD must vanish before it changes sign. This is conveniently expressed by saying that a line M A D с B B segment changes sign when it passes through zero; passing through zero being interpreted as vanishing and reappearing on the other side of the zero-point. Ex. 2. ABCD is a normal quadrangle. Consider the side AD and suppose D to move along the line DA until it comes into the position D'. The segments AD and AD' are opposite in sense, and ABCD' is a crossed quadrangle. .. the crossed quadrangle is derived from А the normal one by changing the sense of one of the sides. Similarly, if one of the sides of a crossed quadrangle be changed in sense the figure ceases to be a crossed quadrangle. Ex. 3. This is an example where segments which are parallel but which are not in line have opposite senses. ABC is a 4 and P is any point within from which perpendiculars PD, PE, PF are drawn to the sides. Suppose that P moves to P'. Then PF becomes P'F', and PF and P'F' being in the same direction have the same sense. Similarly PE becomes P'E', and these segments have the same sense. But PD becomes P'D' which is read in a direction opposite to that of PD. Hence PD and P'D' are opposite in sense. But PD and P'D' are perpendiculars to the same line from points upon opposite sides of it, and it is readily seen that in passing from P to P'the PD becomes zero and then changes sense as P crosses the side BC. Hence if by any continuous change in a figure a point passes from one side of a line to the other side, the perpendicular from that point to the line changes sense. Cor. If ABC be equilateral it is easily shown that PD+PE +PF: =a constant. F Р А E CE And if we regard the sense of the segments this statement is true for all positions of P in the plane. 157o. Product.—The algebraic form of a product of two symbols of quantity is interpreted geometrically by the rectangle having for adjacent sides the segments denoted by the quantitative symbols. This is manifest from Art. 152°, for in the form ab the single letters may stand for the measures of the sides, and the product ab will then be the measure of the area of the rectangle. If we consider ab as denoting a having a as altitude and b as base, then ba will denote the having b as altitude and a as base. But in any o it is immaterial which side is taken as base (138°); therefore ab=ba, and the form satisfies the commutative law for multiplication. Again, let AC be the segment b+c, and AB be the segment C a, so placed as to form the Da(6+c) or AF. Taking AD=b, let DE be drawn || to AB. Then AE and DF are rectF angles and DE=AB=a. DAE is ab, and DF is Dac; Dab+c)=Dab+Dac, and the distributive law is satisfied. А D 6 с a ab ас B E 158°. We have then the two following interpretations to which the laws of operation of numbers apply whenever such operations are interpretable. 1. A single symbol of quantity denotes a line-segment. As the sum or difference of two line-segments is a segment, the sum of any number of segments taken in either sense is a segment. Therefore any number of single symbols of quantity connected by + and signs denotes a segment, as a +b, a-b+c, a-b+(-c), etc. |