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274. By generalizing the use of the fingers in counting, man has made for himself a counting apparatus, which each one carries around in his mind. This counting apparatus, the natural series of numbers, was made from a discrete aggregate, and so will only correspond exactly to discrete aggregates.
275. In a row of shot, we can find between any two, only a finite number of others, and sometimes none at all.
Just so in regard to any two numbers. A row of six shot can be divided into two equal parts; but the half, which is three, we cannot divide into two equal parts: and so in a series of numbers.
276. But in 136 we have shown how any sect whatever may be bisected, and the bisection point is the boundary of both parts. So a line is not a discrete aggregate of points. It is something totally different in kind from the natural series of numbers.
277. The science of numbers is founded on the hypothesis of the distinctness of things. The science of space is founded on the entirely different hypothesis of continuity.
278. Numbers are essentially discontinuous, and therefore unsuited to express the relations of continuous magnitudes.
279. In arithmetic we are taught to add and multiply numbers: we will now show how the laws for the addition and multiplication of these discrete aggregates are applicable to sects, which are continuous aggregates.
THE COMMUTATIVE LAW FOR ADDITION.
280. In a sum of numbers we may change the order in which the numbers are added.
If x and y represent numbers, this law is expressed by the equation
x + y = y + x.
It depends entirely on the interchangeability of any pair of the units of numeration.
281. The sum of two sects is the sect obtained by placing them on the same line so as not to overlap, with one end point in common.
Thus, the sum of the sects a and b means the sect AC, which can be divided into two parts,
AB = α, and BC= b.
282. The commutative law holds for the summation of sects.
a + b = b + a.
AC = C'A' ;
for AC revolved through a straight angle may be superimposed upon C'A', and will coincide point for point.
The more general case, where three or more sects are added, follows from a repetition of the above.
Thus, the commutative law for addition in geometry depends.
entirely on the possibility of motion without deformation.
283. The sum of two rectangles is the hexagon formed by superimposing two sides, and bringing the bases into the same
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.
ab + cd = cd + ab;
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 + %.
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
ad + (bf + cg) = ad + bf + cg.
THE COMMUTATIVE LAW FOR MULTIPLICATION.
288. The product of numbers remains unaltered if the factors be interchanged.
289. The commutative law holds for the rectangle of two
If a and b are any two sects, rectangle ab = rectangle ba,
ab = ba,
for rectangle ab may be so applied to rectangle ba as to coincide 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.
a(b + c) = ab + ac;