In the second place it is to be observed, that the whole coefficient of any power of x in the products of multiplication A, may be reduced to the regular binomial form, established in the 13th article. the whole coefficient of x2 by actual

Thus n

+mn+m

• 2

n+m - 1

=n+m

6

2

the whole coefficient of x3, by actual multiplication becomes n3+m3 −3n2—3m2+3n2m ̧ 3m2 n—6mn+2n+2m

n+m

3

And from the preceding observation it is evident, that we may, in the same manner, reduce the whole coefficient of any other power of x, in the products of multiplication A to the regular binomial form.

(16) But in proceeding, as above, to change the form of the coefficients prefixed to any power of x, in multiplication A, into the regular binomial form, we are not under the necessity of supposing a and m to be whole numbers. The actual multiplications will end in the same powers of n and m, the same combinations of them, and the same numerals, whether we consider n and m as whole numbers, or as fractions.

We are therefore at liberty to suppose n and m to be any two fractions whatever, in the two series multiplied into one another in multiplication A, and the same two fractions will take the place of n and m respectively in the regular binomial series 1+n+mx+n+m. n+m-1 x2+n+m.

2

n+m-2 3

n+m- --3 **+ &c. which expresses the product of the

two series into one another.

⚫ (17) If therefore r be any positive whole number, we can raise

the binomial series 1+

posed power by successive multiplications; or we can express any power of it by supposing the multiplications actually to have been gone through. Thus, calling the last mentioned series the root, if it be multiplied by itself, and if the coefficients in the product be expressed in the regular binomial form, its square will be 1+

-x+&c. Again, if this series be multiplied by the root,

and

and the coefficients in the product be expressed in the regular bino

mial form, the cube of the root will be 1+ x+

&c. Proceeding thus, by multiplying the last found power by the root, in order to find the next higher power, the nth power of 1+

(18) If in the series, which concludes the last article, n be equal

tor, the whole series becomes equal to 1+x.

For in this case

o, and consequently every term

in the series, after the second, becomes equal to o, or vanishes.

Hence it is evident that the rth root of 1+x, or, which is the

3

*+

&c. for this series being raised to the rth power becomes equal to

As by the general principles of involution the ath power of

(20) It is easily proved, by means of the 15th and 16th articles,

that

and n be whole numbers or fractions. For being equal to m―n,

212

this last series becomes 1 − + x + v. 2x2+.

2

3

2

x &c.; and this series being multiplied by 1+nx

expressing their product, by the 15th and 16th articles, is 1+

x1 &c.; and as this equation holds in every possible value

of m, and as, by the general principles of involution, 1+x is equal

We consider it as almost an affront to point out, to any person of moderate mathematical attainments, the similarity, or rather the sameness, of these two proofs. Euler, Art. 5., multiplies the series 1+nx+&c. by the series 1+mx+&c; and he makes the same remark in Art. 6. with regard to the coefficients of the product that is made by Dr. R. in his 16th Article. Again; in Art. 7. Fuler states that, when m and ware. whole numbers, [m] = (1+) and [n] (1+)"; or, according to his own explanation, that the real expanded form

m

for (1x) is 1 + m x + m. x2+ &c.; and for (1+*)*,

(1 + x) is a

2.

2

x2+ &c.; and also that (1+x)(1 + x)m =

m+n: but m and n being whole numbers, († ñ)

whole number, and therefore ( 1 + x) m + n = I + (m + n) x + &c. Consequently (I + mx + &c.) (1 + nx + &c.) = 1 + (m + n) + &c. when m and n are whole numbers; and the form of the product is always the same, (from his remark in Art. 6) whatever m and ʼn are, which in his notation he thus expresses;

[m].[n] = [m+n];

and this is precisely what Dr. R. has done in Art. 15 and 16. In the next Article, (8), Euler shews that [m]a = [am]; that is, if the series 1+mx + m. x2+ &c. be multiplied a

MI
2

times into itself, the resulting series is that which is obtained by substituting am for m in the series 1+mx+ &c.; and therefore if 2m be put i, the series resulting from the multi

i

i

plication of 1++. ( ——— 1 ) x2 + &c. into itself is

2

2

;

what results by putting 2--or i for in the above form:

2

which form, therefore, is r+ix+i.

a whole number, this is the known developed series for

i

(1+*)'; consequently, since i+ * &c. multiplied into

i

itself = (1+x)*, thereforé 1 + - - + &c. =(1+*), which

2

Euler thus expresses: [ + ] = (1 + x) 2 .

In like manner, if 3m be put i, the series 1+ x+ &c. 3 multiplied 3 times into itself, or the cube of the series, will be (1 + i.x + &c.); and if generally ami, then the series +

self, or its ath power, is the series that results when d

i

i. (i-1)

2

a

or is put for; or is the series 1 + ix + x2+ &c.: which, since i is a whole number, is equal to (1+x). Theres føre, since the ath power of 1- *+ &c. is (1+x) —,

If we look to Art. 17. of Dr. R. we shall find a like process conducted on exactly the same principle: 1+++

I

2r

(−1). ×+ &c. multiplied twice, thrice, # times into ité

x

self, produces a series which results from the above by putting

The only difference, that can be pointed out between these two parts, is that Professor R. puts n=r, whereas Euler in fact puts n=mr. If in Dr. R.'s proof we put nmr, then

m

(−1)2+ &c. multiplied n times into it

ހ

n

self, =1+mx + &c. =(1+x)"; or 1+ #x+ &c. = (1+x) »

n

which is the last part of Dr. R.'s 18th article; and which, it is clear from what we have just shewn, he might have had with out going through that part in which m is put = 1, or in which the series 1+mx+ &c. is reduced to 1+x: but the insertion of one unnecessary step is not the sole objection which we have to make against Dr. R.'s mathematics *.

The sameness of principle and process is preserved also in the last parts, in which the theorem is to be proved when the index is negative: but, as we may possibly tire our readers by stripping off the mysterious symbols which concealed the likeness of Euler's proof from Dr. R. we shall vary the proceeding, and clothe the latter in the habiliments of the former.

• If in Euler's proof we put i=1, we have 1+x+ &c. =(1+x)*, and this agrees exactly with Dr. R.: but then in order to obtain

the series for (1+x) we must make another step though whe ther we do or do not introduce an additional step, we see not the elightest difference of principle and method.

Dr..