Imágenes de páginas
PDF
EPUB

a

To prove the rule:-Let be divided by 1, the quotient

a

is evidently; and if it is divided by the dth part of 1, the

ad

quotient will be d times greater than before, or b (123);

again, if is divided by c times b

1 с

or the quotient will be

c times less than the former quotient

ď

[blocks in formation]
[blocks in formation]

From the rule is easily derived the following:(141.) To divide a fraction by an integral quantity, multiply the denominator by it, or divide the numerator when it is divisible."

This is also evident from articles (124) and (125.)

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

(142.) Any proper fraction, with a compound denominator, can, by division, be resolved into an infinite series; for the numerator is a dividend, and the denominator is a divisor related to each other, as the remainder and divisor referred to in article (86); and hence the quotient obtained, by dividing the first term of the numerator by that of the denominator, will not be an integral quantity, but a fraction,

and the process of division will never terminate, so that the quotient will be an infinite series. The examples in Division that come under the observation in article (86), will all by this method produce for a quotient an infinite series. After a few of the terms of the quotient are found, the law of the series will be easily observed, and the succeeding terms can then be obtained without any further division. When a law is thus found, by observing it to hold in every particular instance examined, it is said to be discovered by induction, which is the source of most discoveries in science. When a law is discovered by induction, its truth cannot always be implicitly relied on till it be directly demonstrated.

[blocks in formation]

The law of the series is evident; for after the second

term, if any term be multiplied by, the product will be the succeeding term: this is by induction; but it is evident from the process of the division, that this must be the law of the series.

[blocks in formation]

22

b+x) α-x ( − (a + b) 1/2 + (a + b) 2/3 — &c. (-(a+b)ρ+(a+b)ys

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

y

y2

8... 1 + (x − y) — — (≈—y) &/+(x − y) 2 —...

« AnteriorContinuar »