(1) Let a = b, then x = d = ∞; that is, A can never overtake B; which s also evident from the circumstances; because, if A and B are once d miles apart and travel in the same direction at the same pace, they must always be d miles apart, and can never come together. (2) Let a <b, then x is negative, which signifies that the time of their being together is past. For as A travels more slowly than B, it is evident they cannot at any future time come together, because the farther they go the farther they are apart. But as by looking forward in time the distance between them keeps increasing, so by looking backward (supposing the journey continuous in that direction) that distance continually d diminishes, and hours ago it was 0, that is, A and B were then together. b-a MAXIMA AND MINIMA. (From Bourdon's Alg.) 465. There is a class of problems which require for their solution to determine the greatest or smallest values which an algebraical expression will admit of by the variation of some quantity or quantities contained in it. These problems are called Maxima and Minima Problems. Thus, PROB. 1. Required to divide a given number 2a into two such parts that the product of the two parts may be the greatest possible. Now, that a may have a real value, y cannot be greater than a2, but may be equal to a3, which is therefore its greatest value. Hence, in that case, x = 0, and the two parts of 2 a required are equal to each other. Now, that x may have a real value, (a3 – b3)3y3 must not be less than 4a3b, but it may be equal to it, or y minimum, or least value. 2ab a2 - b3, which is therefore its 466. If the quantity under the radical sign remains positive whatever value be given to y, then the proposed quantity will admit of neither a maximum nor a minimum. Now, whatever value may be given to y, the quantity under the root will be always positive; therefore the proposed expression does not admit of any maximum or minimum. 467. If the quantity under the root be of the form my3+ny+p, then by solving the equation my3 + ny + p = 0, we can find the greatest or least value of y which will permit my2+ny+p to be real, and therefore the required maximum or minimum. Ex. 1. Let a and b be two numbers of which a>b; required the (x+a)(x−b). Ans. Maximum = (a + b) a greatest value of 4ab 2ab ; and x = a - b (a + x) (b+x) Ex. 2. Required the smallest value of Ꮖ Ans. Minimum = (√a+ √b)2; and x = Jab. APPLICATION OF ALGEBRA TO GEOMETRY. 468. The signs made use of in algebraical calculations being general, the conclusions obtained by their assistance may, with great ease and convenience, be transferred from abstract magnitudes to every class of particular quantities; thus, the relations of lines, surfaces, or solids, may generally be deduced from the principles of Algebra, and many properties of these quantities discovered, which could not have been derived from principles purely geometrical. 469. Simple algebraical quantities may be represented by lines. Any line, AB, may be taken at pleasure to represent one quantity a; but if we have a second quantity, b, to represent, we must take a line which has to the former line the same ratio that b has to a. Instead of saying AB represents a, we may say AB = a, supposing AB to contain as many linear units as a contains numeral ones. 470. When a series of algebraical quantities is to be represented on one line, and each of them measured from the same point, the positive quantities being represented by lines taken in one direction, the negative quantities must be represented by lines taken in the opposite direction. Let a be the greatest of these quantities, then a - may, by the variation of a, become equal to each of them in succession. Let AB be the given line, and A the point from which the quantities are to be measured; take AB equal to a; and since a - x must be measured from A, BD must be taken in the contrary direction equal to x, then AD = a; and that a may successively coincide with each quantity in the series, beginning with the greatest positive quantity, must increase; therefore BD, which is equal to a, must increase; and when a is greater than a, BD is greater than AB, and AD', which represents the negative quantity ax, lies in the opposite direction from A. COR. 1. If the algebraical value of a line be found to be negative, the line must be measured in a direction opposite to that which, in the investigation, we supposed to be positive. COR. 2. If quantities be measured upon a line from its intersection with another, the positive quantities being taken in one direction, the negative quantities must be taken in the other. 471. If a fourth proportional to lines representing p, q, r be taken, it will represent ; and if p = 1, it will represent qr; if qr also q and r be equal, it will represent q 472. If a mean proportional between lines representing a and b be taken, it will represent ab, which, when a = 1, becomes √b. Hence it appears that any possible algebraical quantities may be represented by lines; and conversely, lines may be expressed algebraically; and if the relations of the algebraical quantities be known, the relations of the lines are known. 473. The area included within a rectangular parallelogram may be measured by the product of the two numbers which measure two adjacent sides. Let the sides AB, AC of the rectangular parallelogram AD be measured by the lineal quantities a, b, respectively; then a xb will express the number of superficial units in the area, that is, the number of squares it contains, each of which is described upon a lineal unit. For instance, if the lineal unit be a foot, of which AB contains a, and AC contains b, the parallelogram AD contains a × b square feet. For, 1st, if AB, AC be divided into lineal units, and straight lines be drawn through the points of division parallel to the sides, the whole figure is made up of squares, which are equal to each other, and to the square upon the lineal unit; and the number of them is evidently a taken b times, or a b. 2ndly. If AB and AC do not contain the lineal unit an exact number of times, that is, if a and b be fractional, let a = a + and b B+ 1 mn 1 m = n Then take another lineal unit which is -th part of the former ; and by what has been shewn the square described upon the larger unit contains mn × mn of that described upon the smaller. Again, the sides AB, AC respectively contain mna + n, mnß + m lineal units of the smaller kind, and therefore, by the first case, the whole figure contains (mna+n) × (mnẞ+m) square units of the smaller kind; that is, the area 3dly. If the sides AB, AC be incommensurable with the lineal unit, a unit may be found which is commensurable with certain lines that approach as near as we please to AB, and AC, and therefore the product of such lines will represent the area of a rectangle differing from the rectangle AD by a quantity less than any that can be assigned, that is, we may, in this case also, without error express AD by AB × AC. (See Art. 260.) 474. COR. 1. Since, by Euclid, Book 1. Prop. 35, the area of an oblique-angled parallelogram is equal to that of the rectangular parallelogram upon the same base and between the same parallels, therefore area of any parallelogram = base × altitude. 475. COR. 2. Also, since by Euclid, Book 1. Prop. 41, the area of a triangle is half that of the parallelogram upon the same base and between the same parallels, therefore 476. COR. 3. Since any rectilineal figure may be divided into triangles, its area may be found by taking the sum of all the triangles. 477. The solid content or volume included within a rectangular parallelopiped may be measured by the product of the three numbers which measure its length, breadth, and height. Let the base of the parallelopiped be divided into its component squares, as in the preceding Proposition, and through each of the parallels suppose planes drawn at right-angles to the base; and let the same thing be done with one of the faces adjacent to the base. Then it is evident that the whole figure is divided into a certain number of equal cubes, each cube having for its face one of the squares described upon the lineal unit; (that is, if the lineal unit be a foot, each of these cubes will have its length, breadth, and height equal to a foot, and is called a cubic foot). Now the number of these cubes is manifestly equal to the number of squares in the base taken as many times as there are lineal units in the height; therefore content or volume = base × height = length × breadth × height. COR. Any three of these quantities being given, the fourth may be |