respectively, and therefore, by the first case, the whole figure contains (mna+ n) (mnß + m) square units, of the smaller kind; that is, the area

1

- m3n (a + }{3+} of the smaller units,
B+
m

=

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. 248.)

XVII.

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

XVIII. 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

XIX. COR. 3. Since any rectilineal figure may be divided into triangles, its area may be found by taking the sum of all the triangles.

XX. 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 x height

COR.

=

Any three of these quantities being given, the fourth may be found. Thus, if C be the content, 7 the length, b the breadth, and h the height, we have

MAXIMA AND MINIMA. (From Bourdon's Alg.)

XXI. 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 2 a into two such parts that the product of the two parts may be the greatest possible.

Now, that may have a real value, y cannot be greater than a2, but may be equal to a2, which is therefore its greatest value. Hence, in that case, 0, and the two parts of 2a x = required are equal to each other.

Now, that a may have a real value, (a2 – b2)2y2 must not be

less than 4a2b2, but it may be equal to it, or y =

is therefore its minimum, or least value.

2ab

which

a2 - b29

XXII. 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.

Ex. Required to determine whether

of either a maximum or 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.

XXIII. If the quantity under the root be of the form my2+ny+p, then by solving the equation my2+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 maxi

mum or minimum.

Ex. 1. quired the greatest value of

Let a and b be two numbers of which a > b, re

(x+a) (x − b)

(a + x) (b + x)

Ex. 2. Required the smallest value of

Ans. Minimum (√a+√√/b)2; and x = √ ab.

SINGLE AND DOUBLE POSITION.

XXIV. The Arithmetical Rules called "Single Position," and "Double Position," may be thus investigated.

(1) The Rule of Single Position' is applied to those cases only, in which the required quantity is some multiple, part, or parts, of some other given quantity; that is, if a represent the required value, a and b known quantities, the cases for Single Position' are such as produce an equation of the form

Thus, if it be required to find such a value x that a × x = b, suppose s to be the value, and instead of b, we find a x s = b', then we have

which points out the Rule:—namely, Suppose some value (s) to be the one sought, and having operated upon it as the question directs, let the result (b') be noted. Then the true value is equal to the true result (b) divided by the false one (b') and multiplied by the supposed value (s).

Ex. Find the number which being added to the half and fourth of itself will produce 14.

Suppose 12 the number, then

12+ of 12 + 1 of 12 is 21;

(2) The rule for Double Position' is applied to those cases in which the required quantity is not a multiple, part, or parts, of a given quantity, but furnishes an equation of the form

ax + b = cx + d.

By transposition this equation becomes

(ac) x + b d = 0.........(1).

Now suppose to be the value of x, which by substitution does not satisfy the equation, but gives

(ac) s + b - d = C.........(2),

then subtracting (1) from (2),

(ac) (8x) = e.

Again, supposes to be the value of x, which by substitution and subtraction, as before, gives

which proves the common rule, namely, Make two suppositions (s and s') for the required quantity; treat each of them in the manner pointed out by the question; and note the errors (e and e'); then the required quantity will be found by dividing the difference of the products es', e's by the difference of the errors e, e'.

Ex. What number is that which, upon being increased by 10, becomes three times as great as it was before?