TRACT 24.

©F NUMBERS.

463

3. For the Reciprocals of Numbers.

The reciprocals of the natural numbers a, a + 1, a + 2,a + 3, &c, are denoted by the fractions — - ——, ~—&c, where a is any integer number to commence with; whichreciprocals, with their several orders of differences here follow.

Recips.

1st Diffs.

2d Diffs.

3tl Diffs.

i

a

1

1

a.a+ 1

1 . 2

1.2.3

a+ 1

1

1

a.a +1.a+2

1 . 2

a . a + 1 . a+ 2.a + 31.2.3

a + 2

1

Q + l«ft + 2 M + 3

a + 1 .a+2.a+3.a + 4>

1

a+2 a + 3

« + 3

1st Dif.

2a+ 22a + 42a+ 6

2d 5);

2

2

Here, if we would employ only the column of first differ-ences, by actually multiplying the terms in their denomi-nators, these, with their two orders of differences, will be asfollow.

Where the first differences are D^noms.in arithmetical progression, and ar + athe 2d differences equal, viz, the a 2 +3a+ 2

constant number 2. Hence the a z + 5a-\- 6

series of denominators will be very + 7 a J r ! 2

soon constructed, by two easy additions, the firstof which is bythe constant number 2. So, for instance, it a be =1000, themthe first differences, and their denomina- 1 st Dif. Denoms.tors, will be thus : Where the column of 2002 1001000

first diffs. increases always by the number 2004 1003002

2, and the column of denominators is 2006 1005006

constructed by adding the several first differences. Thesedenominators are so large, that a very few figures in theirquotients, will be sufficient to form, by one addition foreach, the original column of reciprocals, to a great manyplaces of figures. And these reciprocals will be verified andcorrected at every 10th number; for any reciprocal whosedenominator ends with a cipher, will have the same signifi-