Supposing now, that the notation is completely understood, we will proceed to the reduction of decimal fractions.
To reduce a vulgar fraction to a decimal : set down the numerator of the fraction, with a point upon the right side of it, add as many ciphers in succession, towards the right hand, as may be thought necessary ; then, if the denominator consist of one single figure, or of two, not exceeding 12, draw a horizontal line below the row of figures so set down, and a vertical line upon the left side of the left-hand figure: set down the denominator upon the left of this line, proceed as in short division. placing a point under the other, then if the succeeding figures towards the right begin under the first figure after the point, these figures will be the decimal, but if not, the number corresponding to the upper row must be made out, by adding a cipher to the left hand.
Example 1. Required the decimal of I.
4) 1.00 .25 the decimal required.
Example II. Required the decimal of -L.
2) 1.0 .5 the decimal required.
Example III. Required the decimal of 1.
4) 3.00 .75 the decimal required.
The reader who wishes to employ decimals in his ealeula. Lions, should have the decimals .25, .5, .75, of 1, fixed on his memory.
Example IV. Required the decimal of 1 inch in terms of a foot. Here 1 inch is the twelfth part of a foot, therefore the vulgar fraction is 12) 1.00000 .0S333 Example V. Required the decimal of 2 inches. Now 2 inches is A- or 1; therefore 12) 2.00000 or 6) 1.00000 .16666 .16666E.rample VI. What is the decimal of 3 inches ? 3 inches is equal to or I, therefore the decimal will be .25, as above.
Example VII. Required the decimal of 4 inches. 4 inches is equal to = therefore, 12) 4.00000 or 3) 1.00000 .33333 .33333In this manner, the decimals for every number of inches under twelve are to be found, as the following table shows : The decimal of 1 inch = .08333 2 = .166663 4 = .33333 5 = .4166G 6 = .5 7 = .5S333 8 = .66666 9 =.75 10 = .8333311 = .91666 In most practical cases, three figures of decimals will be found sufficient.
When inches, seconds, thirds, &c. are to be reduced to a decimal, the best method is to reduce the feet, inches, &c. to the last denomination, then divide as often by 12 in succession is there are denominations, and the last quotient will be the decimal required.
Example I. Required the decimal of 9 firsts, or inches, mud 6 seconds.
Now here are two denominations, therefore . 912 12) 114 seconds in the whole.
12) 9.57.9160 the decimal required.
And thus for any other number of denominations whatever.
If each foot of our measuring-rules for taking the lineal dimensions were divided into ten parts, instead of twelve, and each of these ten parts again into ten others, we should have tio occasion for reduction of decimals, as the rule itself would give the decimal. In most cases we should not then have occasion to work with more than two places of decimals; the tenth part of the tenth part, that. is, the hundredth part of a f /of_ is very nearly equal to the eighth part of an inch, or the ninety-sixth part of a foot, being only a small matter less than the eighth of an inch. Were measuring-rules thus divided. the work by decimals be much shorter than any other whatever. The principal reaseat of operations in measuring by dceinuds being longer than duodecimals, arises from the necessity of reducing the duode cimals to decimals, and this in many cases cannot be done with the same accuracy, without having four or five denomi nations. In practical cases there are never more than three places of duodecimals, and if rules were divided decimally, there not be more than three places.
A decimal part ()la foot being given, to find its equivalent in duodecimals.
Multiply the decimal by 12, cut off as many decimals from the product as there are places of figures in the multiplicand, from the right-hand to the left, and the figure or figures remaining on the left, if any, will show the number of inches : multiply the number of decimals so cut off, if any, again by 12. and cut otT as many figures from the left-hand of the new product as there are decimal places in the multiplicand, and the remaining figures on the left will show the seconds, if any. Proceed in this manner as often as there is a remainder, or as often as may be thought necessary to obtain a sufficient degree of accuracy, and the places cut off will be equivalent to the given decimal, if no remainder, and very nearly so if there is, but in this ease something less.