283. The sum of two rectangles is the hexagon formed by superimposing two sides, and bringing the bases into the same

line.

C

A

G b

Thus, if two adjacent sides of one are a and b, and of the other c and d, the sum of the rectangles ab and cd is ABCDFGA.

284. The commutative law holds for the addition of rectangles; that is, the sum is independent of the order of summation.

for ABCDFGA turned over may be superimposed upon A'B'C'D'F'G'A', and will coincide with it.

Since, by 263, we can describe on a given base a rectangle equivalent to a given polygon, the more general case, where three or more rectangles are added, follows from a repetition of the above.

THE ASSOCIATIVE LAW.

285. In getting a sum of numbers, we may add the numbers together in groups, and then add these groups.

If we use parentheses to mean that the terms enclosed have been added together before they are added to another term, this law may be expressed symbolically by the equation

x + (y + z) = x + y + z.

286. The associative law holds for the summation of sects. a + (b + c) = a + b + c = AD.

287. The associative law holds for the summation of rect

THE COMMUTATIVE LAW FOR MULTIPLICATION.

288. The product of numbers remains unaltered if the factors be interchanged.

xy = yx.

289. The commutative law holds for the rectangle of two

or

If a and b are any two sects, rectangle ab = rectangle ba,

for rectangle ab may be so applied to rectangle ba as to coin

cide with it.

THE DISTRIBUTIVE LAW.

290. To multiply a sum of numbers by a number, we may multiply each term of the sum, and add the products thus obtained.

x(y + z) = xy + xz.

291. The distributive law holds when for numbers and products we substitute sects and rectangles.

for if we add the rectangle ab to the rectangle ac, so that a side a in the one shall coincide with an equal side a in the other, the sum makes a rectangle whose base is b + c and whose altitude is a; that is, the rectangle a(b + c).

In the same way, by adding three rectangles of the same altitude, we get

a(b + c + a) = ab + ac + ad.

We

may state this in words as follows:

If there be any two sects one of which is divided into any number of parts, the rectangle contained by the two sects is equivalent to the rectangles contained by the undivided sect and the several parts of the divided sect.

292. If b + c = a, then

ab + ac = a(b + c) = aɑ = a2.

a

Therefore

If a sect be divided into any two parts, the rectangles contained by the whole and each of the parts are together equivalent to the square on the whole sect.

If a sect be divided into any two parts, the rectangle contained by the whole and one of the parts is equivalent to the rectangle contained by the two parts, together with the square on the aforesaid part.

294. The rectangle of two equal sects is a square, and (a + b) is only a condensed way of writing (a + b)(a + b). But, by the distributive law,

(a + b) (a + b) = (a + b)a + (a + b)b.

By the commutative law,