23. To inscribe any regular polygon in a given circle. Divide any diameter, as AB, into so many equal parts as the polygon is required to have sides; from A and B as centres, with a radius equal to the diameter, describe ares cutting each other in C; draw the line CD through the second point of division on the diameter e, and the line D B is one side of the polygon required. 25. To form a square equal in area to a given triangle. 26. To form a square equal in area to a given rectangle. tangle; bisect the line in e, and describe the semicircle AD B; then from A with the breadth, or from B with the length, of the rectangle, cut the line AB at C, and erect the perpendicular CD, meeting the curve at D, and CD equal a side of 27. To find the length for a rectangle whose area shall be equal to that of a given square, the breadth of the rectangle being also given. Let A B CD be the given square and DE the given breadth of pe rectangle; continue the line B C to F, and draw the line DF; also continue the line DC to g, and draw the line Ag parallel to DF; from the intersection of the lines at g, draw the line gd parallel to DE, and Ed parallel to Dg; then EDdg is the rectangle as required. 28. To bisect any given triangle. D E Suppose A B C the given triangle; bisect one of its sides, as A B in e, from which describe the semicircle Ar B; bisect the same in r, and from B, with the distance Br, cut the diameter AB in ; draw the line y parallel to A C, which will bisect the triangle as required. 29. To describe a circle of greatest diameter in a given triangletulan dala Bisect the angles A and B, and draw the intersecting lines ting each other in DD, BD, cut, from D as centre, with the distance or radius D C, describe the circle Cef, as required. F B 30. To form a rectangle of greatest surface, in a given triangle. Let A B C be the given triangle; bisect any two of its sides, as A B, B C, in e and d; draw the line ed; also, at right angles with the line e d, draw the lines e p, dp, and ep pd is the rectangle required. с R e 6. Tín, 7. Lead. B STRENGTH OF WOOD. All woods are from 7 to 20 times stronger transversely than lon gitudinally. They become stronger both ways when dry. 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 one-hundredth 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 pecu liar 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 converting 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. Ex. 2. value. Reduce the fraction of an inch to a decimal of equal 1.000 125, the decimal required. 8 Ex. 3. What is the decimal equivalent to of a gallon? 7.00.0 875, the decimal equivalent. Ex. 4. Required the fractional value of the decimal .40625 of an inch. 100000 and of an inch, the value required. Ex. 5. What is the fractional value of '625 of a cwt.? 625 Multiply by 4 qrs 14.0002 quarters and 14 lbs., the value required. Ex. 6. Ascertain the fractional value of 875 of an imperial gallon. 1·000=3 quarts and 1 pint, the value rerequired. Ex. 7. What is the fractional value of 525 of a £. sterling? Multiply by 20 sh. × 12 pence *525 10.500 12 6.000 10 shillings and 6 pence, the value quired. Independent of the mark or dot which distinguishes between integers and decimals, the fundamental rules-viz. Addition, Subtraction, Multiplication, and Division are in all respects the same as in Simple Arithmetic; and an example in each, illustrative of placing the separating point, will no doubt render the whole system | sufficiently intelligible, even to the dullest capacity. Ex. 1. Add into one sum the following integers and decimals: 16.625; 114; 20-7831; 12.125; 8'04; and 7′002. Observe that the number of figures in the product from the right hand, accounted as decimals, are equal to the number of decimals in the multiplier and multiplicand taken together. Note. The operation might be still continued, so as to reduce the quotient to a degree of greater exactitude; but in practice it is quite unnecessary, being even now reduced to a measure of greater nicety than is commonly required. |