By transposing the terms in the expressions marked 1, we have the following values of A, B, C, &c. By which the co-efficients of an equation may be found, from the sum of its roots, the sum of their squares, the sum of their cubes, &c. Ex. 1. Required the sum of the roots, the sum of their squares, and the sum of their cubes, in the equation x2-10x3 +35x2 - 50x—24=0. S, 103+(3x-10x35)-(3x-50) 100. 2. Required the terms of the biquadratic equation in which S, 1, S, 39, S,-89, and the product of all the roots after their signs are changed is -30. Ans. x1 —x3 — 19x2+49x — 30—0.* *See Note V. SECTION XXI. APPLICATION OF ALGEBRA TO GEOMETRY.* ART. 504. It is often expedient to make use of the algebraic notation, for expressing the relations of geometrical quantities, and to throw the several steps in a demonstration into the form of equations. By this, the nature of the reasoning is not altered. It is only translated into a different language. Signs are substituted for words, but they are intended to convey the same meaning. A great part of the demonstrations in Euclid, really consist of a series of equations, though they may not be presented to us under the algebraic forms. Thus the proposition, that the sum of the three angles of a triangle is equal to two right angles, (Euc. 32. 1.) may be demonstrated, either in common language, or by means of the signs used in algebra. Let the side AB, of the triangle ABC, (Fig. 1.) be continued to D; let the line BE be parallel to AC; and let GHI be a right angle. The demonstration, in words, is as follows: 1. The angle EBD is equal to the angle BAC, (Euc. 29. 1.) 2. The angle CBE is equal to the angle ACB. 3. Therefore, the angle EBD added to CBE, that is, the angle CBD, is equal to BAC added to ACB. 4. If to these equals, we add the angle ABC, the angle CBD added to ABC, is equal to BAC added to ACB and ABC. *This and the following section are to be read after the Elements of Geometry. 5. But CBD added to ABC, is equal to twice GHI, that is, to two right angles. Euc. 13. 1. 6. Therefore, the angles BAC, and ACB, and ABC, are together equal to twice GHI, or two right angles. Now, by substituting the sign +, for the word added or and, and the character, for the word equal, we shall have the same demonstration, in the following form. 3. Add the two equations EBD+CBE=BAC+ACB 4. Add ABC to both sides CBD+ABC=BAC+ACB+ABC 5. But by Euclid 13. 1. CBD+ABC=2GHI 6. Make the 4th & 5th equal BAC+ACB+ABC=2GHI. By comparing, one by one, the steps of these two demonstrations, it will be seen, that they are precisely the same, except that they are differently expressed. The algebraic mode has often the advantage, not only in being more concise than the other, but in exhibiting the order of the quantities more distinctly to the eye. Thus, in the fourth and fifth steps of the preceding example, as the parts to be compared are placed one under the other, it is seen, at once, what must be the new equation derived from these two. This regular arrangement is very important, when the demonstration of a theorem, or the resolution of a problem, is unusually complicated. In ordinary language, the numerous relations of the quantities, require a series of explanations to make them understood; while, by the algebraic notation, the whole may be placed distinctly before us, at a single view. The disposition of the men on a chess-board, or the situation of the objects in a landscape, may be better comprehended, by a glance of the eye, than by the most laboured description in words. 505. It will be observed, that the notation in the example just given, differs, in one respect, from that which is generally used in algebra. Each quantity is represented, not by a single letter, but by several. In common algebra, when one letter stands immediately before another, as ab without any character between them, they are to be considered as multiplied together. But in geometry, AB is an expression for a single line, and not for the product of A into B. Multiplication is denoted, either by a point, or by the character X. The product of AB into CD, is AB CD, or AB × CD. 506. There is no impropriety, however, in representing a geometrical quantity by a single letter. We may make b stand for a line or an angle, as well as for a number. If, in the example above, we put the angle 1 a+c=g=b+d g+h=b+d+h g+h=2 This notation is, apparently, more simple than the other; but it deprives us of what is of great importance in geometrical demonstrations, a continual and easy reference to the figure. To distinguish the two methods, capitals are generally used, for that which is peculiar to geometry; and small letters, for that which is properly algebraic. The latter has the advantage in long and complicated processes, but the other is often to be preferred, on account of the facility with which the figures are consulted. 507. If a line, whose length is measured from a given point or line, be considered positive; a line proceeding in the opposite direction is to be considered negative. If AB, (Fig. 2.) reckoned from DE on the right, is positive; AC on the left is negative. A line may be conceived to be produced by the motion of a point. Suppose a point to move in the direction of AB, and to describe a line varying in length with the distance of the point from A. While the point is moving towards B, its distance from A will increase. But if it move from B towards C, its distance from A will diminish, till it is reduced to nothing, and will then increase on the opposite side. As that which increases the distance on the right, diminishes it on the left, the one is considered positive, and the other negative. See arts. 59, 60. Hence, if in the course of a calculation, the algebraic value of a line is found to be negative; it must be measured in a direction opposite to that which, in the same process, has been considered positive. (Art. 197.) 508. In algebraic calculations, there is frequent occasion for multiplication, division, involution, &c. But how, it may be asked, can geometrical quantities be multiplied into each other? One of the factors, in multiplication, is always to be considered as a number. (Art. 91.) The operation consists in repeating the multiplicand, as many times as there are units in the multiplier. How then can a line, a surface, or a solid, become a multiplier? To explain this, it will be necessary to observe, that whenever one geometrical quantity is multiplied into another, some particular extent is to be considered the unit. It is immaterial what this extent is, provided it remains the same, in different parts of the same calculation. It may be an inch, a foot, a rod, or a mile. If an inch is taken for the unit, each of the lines to be multiplied, is to be considered as made up of so many parts, as it contains inches The multiplicand will then be repeated, as many times, as there are units in the multiplier, if for instance, one of the lines be a foot long, and the other half a foot; the factors will be, one 12 inches, and the other 6, and the product will be 72 inches. Though it would be absurd to say that one line is to be repeated, as often as another is long; yet there is no impropriety in saying, that one is to be repeated as many times, as there are feet or rods in the other. This, the nature of a calculation often requires. 509. If the line which is to be the multiplier, is only a part of the length taken for the unit; the product is a like part of the multiplicand. (Art. 90.) Thus, if one of the factors is 6 inches, and the other half an inch, the product is 3 inches. 510. Instead of referring to the measures in common use, as inches, feet, &c. it is often convenient to fix upon one of the lines in a figure, as the unit with which to compare all the others. When there are a number of lines drawn within |