circle bf, draw at right angles, and through the centre, the line ce; draw also the line or chord fg, cutting the line vn in g; then, with the distance vg, describe the quadrant irs; bisect the arc in r, from which and parallel with vn, draw the line rt; draw also the lines ft, gt, which determines the breadth of the ellipsis or projected form of the circle, and which may be drawn as described (page 5, fig. 12): hence, all the other lines being so distinctly described by mere inspection of the diagrams, further explanation is un

necessary.

Fig. 2 is a projected representation of an undershot water wheel, of which A is the plan, vn the line of intersection, and B the diameter and breadth, laid at a proper angle or inclination, as determined by the principles of the diagrams: the breadth of the ellipsis is found and formed as there described, and the lines in the illustration, or fig. 2, whereby to obtain the proper angles for projection, render sufficiently explicit the proper mode by which the representation is effected.

DECIMAL ARITHMETIC.

DECIMAL ARITHMETIC is the most simple and explicit mode of performing practical calculations; on account of its doing away with the necessity of fractional parts in the fractional form, thereby reducing long and tedious operations to a few figures arranged and worked in all respects according to the usual rules of common arithmetic.

Decimals simply signify tenths; thus, the decimal of a foot is the tenth part of a foot, the decimal of that tenth is the hundredth of a foot, the decimal of that hundredth is the thousandth of a foot, and so might the divisions be carried on and lessened to infinity; but in practice it is seldom necessary to take into account any degree of less measure than a onehundredth part of the integer or whole number. And, as the entire system consists in supposing the whole number divided into tenths, hundredths, thousandths, &c., no peculiarity of notation is required, otherwise than placing a mark or dot, to distinguish between the whole and any part of the whole: thus 34.25 gallons signify 34 gallons 2 tenths and 5 hundredths of a gallon; 11.04 yards signify 11 yards and 4 hundredths of a yard; 16 008 shillings signify 16 shillings and 8 thousandth parts of a shilling: from which it must appear plain, that ciphers on the right hand of decimals are of no value whatever; but placed on the left hand, they diminish the decimal value in a tenfold proportion, for 6 signify 6 tenths; 06 signify 6 hundredths; and 006 signify 6 thousandths, of the integer or whole number.

REDUCTION.

Reduction means the construing or changing of

vulgar fractions to decimals of equal value; also finding the fractional value of any decimal given.

Rule 1. Add to the numerator of the fraction any number of ciphers at pleasure, divide the sum by the denominator, and the quotient is the decimal of equivalent value.

Rule 2. Multiply the given decimal by the various fractional denominations of the integer or whole number, cutting off from the right hand of each product for decimals a number of figures equal to the given number of decimals, and thus proceed until the lowest degree, or required value, is obtained.

Ex. 1. Required the decimal equivalent, or decimal of equal value, to of a foot.

12

25, the decimal required.

Ex. 2. Reduce the fraction of an inch to a decimal of equal value.

Ex. 3. What is the decimal equivalent to of a gallon?

7.000

8

=875, the decimal equivalent.

Ex. 4. Required the fractional value of the decimal

1.00000 and of an inch, the value required.