TRACT 24.

OF NUMBERS.

463

3. For the Reciprocals of Numbers.

The reciprocals of the natural numbers a, a -f- 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.

3d Diffs.

1

a

i

i

a. a+ 1

1 . 2

1.2.3

a + 1

1

a . a +1 .a + 2

a.a+1 .a+2.a + 3

1

a+ 1 .a + 2

1 . 2

1.2.3

a+2

1

a+ l.a + 2.a + 3

a+1 .a + 2.a + 3.a + 4<

1

a+2 a+3

8 + 3

1st Dif.

2a+ 22 a.+ 42a+6

sd D-

2

2

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

Where the first differences are Denoms.in arithmetical progression,, and ar + a

the 2d differences equal, viz, the a z + 3a-J- 2

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

series of denominators will be very a z -\-la-\- 12

soon constructed, by tw'o easy additions, the first of which is bythe constant number 2. So, for instance, if a be =1000, thenthe first differences, and their denomina- 1st 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 manj'places 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-