bers to n-1, inclusive, it is plain that the number n exactly measures some one of the numbers in the scale m(m—1), (m—2), (m—3), &c. Hence, if we make, successively, each number in the scale 1, 2, 3, 4, &c. equal to n, we shall show, m(m - 1) (m 2) (m

as above, that the formula

4

3)

consisting of any number of terms, is a whole uumber.

&c.,

This formula the student will recognize as the formula for finding the numeral coefficients in the Binomial Theorem of Newton.

Demonstration of the Binomial Formula.

448. Let m be the index of the power to which a binomial is raised; the coefficients of any term which is preceded by n terms is the number of combinations of m things, taken n by n. This law or property M. Bourdon, in his Elémens D'Algèbre, has shown as follows:

"The more easily to discover the law of the developement of the mth power of the binomial x+a, we shall begin by observing the law which exists in the product of several binomials x+a, x + b, x + c, x+d... having a common first term, and their second terms different. The object of this artifice is to prevent the reduction of the similar terms.

x + a
x+b

+cd

"Having performed the multiplications according to the usual rules, we discover in the three preceding products the following law:

"1. With regard to the exponents or indices, the index of x

is, in the first place, equal to the number of binomials multiplied. This index then diminishes, by a unit at a time, in each successive term, to the last, where it is equal to nought.

"2. With respect to the coefficients of the different powers of x, the coefficient of the first term is a unit; the coefficient of the second term is equal to the sum of the second terms of the binomials; the coefficient of the third term is equal to the sum of the different products, or combinations of the same second terms taken 2 by 2; the coefficient of the fourth term is equal to the sum of the different products or combinations, 3 by 3. Pursuing the analogy, we may say that the coefficient of any term, which has n terms before it, is equal to the number of different products or combinations of the second terms of the binomials, taken n by n."

term.

Now the number of second terms is equal to the number of binomials, or index of the power of their leading quantity x. Hence, we infer that, when a binomial is raised to any power m, the index of the power, or m, is the coefficient of the second The coefficient of the third term is the number of combinations of the same number m, taken 2 and 2. But m(m 1). ; that is, it is the coefficient of the second term, multiplied by the index of the leading quantity in that term, and divided by the number which shows the number of terms to that place, or number preceding the third term. The coefficient of the fourth term being the combinations of the m(m — 1) (m - 2) same number m, taken 3 by 3, is

this is

1

2 ·

1

2

3

; that is,

it is the coefficient of the third term, multiplied by the index of the leading quantity in that term, and divided by the number showing the number of terms preceding the fourth. Wherefore, in general, the coefficient of any term is found by multiplying the coefficient of the term preceding it, by the index of the leading quantity in that term, and dividing the product by the number which shows the number of terms preceding the term the coefficient of which is sought.

-

449. The number of combinations of which m things are susceptible is the same when the m things are taken 1 by 1, as when taken (m − 1) by (m − 1); when they are taken 2 by 2, as when they are taken (m2) by (m − 2); and, in general, when taken n by n, as when taken (mn) by (mn), n being less than m.

Let n=1; then (m · — n + 1) = m; that is, m is the number of combinations, when taken 1 at a time.

Let n = m

-1; then (m- n + 1)

- W =

(m

− 1)

+1, or mm+1+1=2. Wherefore, the formula m(m—1) (m2) (m3)...2

becomes

1 2.(n-3)(n-2) (n − 1)n' m|(n) (n − 1)(n-2) (n − 3), &c. 2

1.2.(n-3) (n − 2) (n − 1)

That is to say, when m things are taken (m—1) at a time, the number of combinations is equal to m, the same as when taken 1 at a time.

shows that the number of combinations is the same when m things are taken 2 at a time as when taken m 2 at a time. Hence, we may infer that the number of combinations is the same when m things are taken n at a time as when (m — n) at a time.

450. Also, the number of combinations is greatest when

m

n = that is, when n is the half of m.

For, as the last term in the numerator of the fractional formula is (m n+1), and the last term in the denominator is n, it is plain that as long as (m―n+1) is greater than n, the greater n is; that is to say, (444,) the greater the number of terms, the greater will be the number of combinations; but, when (m n+1) becomes less than n, the greater the number of terms, the less must be the result, or number of combinations, because the series which constitutes the numerator of the formula is decreasing, while that of the denominator is increasing. Now it is plain that (m-n + 1) will

m

m

-be greater than n, when n = but when n + 1,

that is, when n is a unit greater than the half of m, then

and is, therefore, less than n. Wherefore, the number of combinations is greatest when n is the half of m.

451. When m is odd, and consequently its half fractional,

m

n, being integral, cannot equal; but, taking ʼn as near n

as possible, m

2

n must equal n+1, or n·

m

n = n+1, then m

m

2

1. First, if

n+1: =n+2. Again, if 1+ 1, or n; and

· 1, then m— · n + 1 = n⋅

m

--

as these are both equally near the number of combinations

For example, let us take m equal to the odd number 9;

m 9

then = = 2 2 integers, to 4, n will be 4 or 5; and, as these are equally near, the number of combinations must, in either case, be the same. First, let n= 4; then m · n + 1

4. Now, taking n as near as we can, in

Now, applying this to the coefficients of the binomial, we may observe that when the power is odd, the number of terms is even, and we always have two middle terms, which are alike, and the greatest in the scale. But when the power is even, and, consequently, the number of terms odd, we have only one middle term, the coefficient of which is the greatest.

Arithmetical Proportion.

452. The difference between two quantities is called their arithmetical ratio. Thus 3 is the arithmetical ratio of 5 to 2, or of 7 to 4. An arithmetical ratio is written thus: 5.2, and is read 5 is to 2. An arithmetical proportion is formed of two equal arithmetical ratios.

When four quantities a, b, c, d are such that a or ba d-c, they, and only such, are said to be in arithmetical proportion. The proportion is written thus: a. b:c. d, and is read a is to b, as c is to d.

In an arithmetical proportion, the sum of the extremes is equal to the sum of the meuns: and this, which is called the fundamental property, may be thus demonstrated: Let a.bc.d be a general representation of every arithmetical proportion. First, let the antecedents, a and c, be greater than their consequents b and d; and let a-b -b=c— q, because a— Hence, a = b+q, g+d; consequently, a +d=b+g+d, and b+c=b+q+d; wherefore, a + d = b+c.

=

с - d and c =

- d.

9, then

Again, let the consequents b and d be greater than their antecedents, and let b—a=q; then d—c=q. Also, b=a+q, and d=q+e; and hence, a+da+q+c, and b+c=a+q+c; consequently, a +d=b+c, as before. Thus, in the proportion 5.2:7.4, we have 5 +4 =7+2. Wherefore, in every arithmetical proportion the sum of the extremes is equal to the sum of the means.

453. From this fundamental property, it is evident, that we may change the places of the means or those of the extremes; also, that we may take the means for the extremes, and the extremes for the means, without disturbing the proportion, because, in all these changes, the sum of the extremes will be equal to that of the means.

454. When in an arithmetical proportion, the two middle terms are alike, the proportion is said to be continual. Thus, 2.5: 5.8, or a.bb.c, is a continual proportion; and is sometimes written without repeating the middle term, thus:

2.5.8, ora.b.c. The preceding sign is used, not only to distinguish the proportion from a common multiplication, but to signify the repetition of the middle term.

The middle term is called a mean proportional, or arithmetical mean, between the other two. In the above proportions, 5 is an arithmetical mean between 2 and 8; and b, an arithmetical mean between a and c.

455. As the sum of the extremes is equal to that of the means, it is evident that we shall find the arithmetical mean between any two numbers or quantities by taking half their

sum. Thus, to find a mean between 5 and 8, we have

=

13

=6; therefore, 5.6.8, or÷8.64.5.

2

5+8

2

The difference which reigns in the proportion between term and term, is called the ratio.