464'

HOOTS AND RECIPROCALS

TRACT 24.

1 st Difis.

Reciprocals.

Nos.

000000999

■000999001

1001

000000997

•000998004

1002

000000995

•000997009

1003

000000993

•000996016

1004

cant figures as the reciprocal of its 10th part, which, it issupposed, has been before found.

The above first differences and denominators will be suffi-cient to construct the table of reciprocals, commencing withthe number 1000, as far as 9 places of decimals, the constant2d difference bcin<r 2 inthe 9th place, for a con-siderable way. Thus, di-viding 1 by the severaldenominators above setdown, gives for their quotients the annexed column of firstcliffs, and thence their annexed reciprocals, &c.

But if a table of reciprocals be desired to a greater numberof decimals, we might take in, and employ, the column of2d differences also ; by which means we should obtain theseries of reciprocals to 12 places of decimals. And so on, forstill more figures.

From the last two or three Tracts, may be constructed, ormay be easily continued further, such tables as here nextfollow, of the reciprocals, squares, cubes, and roots of thenatural series of integer numbers ; the use of which is evi-dently to shorten the trouble of arithmetical calculations.The structure of the table is evident: the first column con-tains the natural series of numbers, from 1 to-1000 ; the 2dthe squares of the same ; the 3d the cubes ; the 4th the reci-procals ; the 5th the square roots; and lastly the cube rootsof the same. The decimals, in the columns of reciprocalsand roots, are all set down to the nearest figure in the lastdecimal place; that is, when the next figure, beyond the lastplace set down in the table, came out a 5 or more, the lastfigure was increased by 1 ; otherwise not; except in the re-petends, which occurred among the reciprocals, where thereal last figure is always set down. Those reciprocals whichin the table have less than seven places of figures, are suchas terminate, and are complete within that number, havingnothing remaining; such as - 5 the reciprocal of 2, '25 thereciprocal of 4, See. The manner and cases of applying these