874

THEORY AND PRACTICE OE ENGINEERING.

Book II.

I'ig. 1140.

fractions or decimal quantities, which, when added or taken away, often materially affectthe whole quantity.

To measure the Circle —This cannot be a ♦ A

done in any other manner than by ap-proximation, as we cannot find the quadra-ture of the circle, or the superficial con-tent of a square, which shall containprecisely that of a given circle. If atriangle A has three equal sides, each 14feet 8 inches long, its circumference willbe 44 feet: if the square B has each ofits four sides 11 feet long, its circum-ference will be 44 feet; and if the circle Chas its circumference 44 feet, these threefigures will be isoperimetrical. If youmeasure the triangle A and the squareB by the rules given already, you willfind that the square contains 121 feetsuperficial, the triangle 45 feet 4 inches,and the circle 154 superficial feet. Ofall w r ho have written on the subject, nonehave given rules which approximatenearer to truth than Archimedes ; hestates the circumference of a circle is 3jthe diameter, that is to say, the circum-ference A B C D will be 22 feet if itsdiameter AC is 7 feet, so that thediameter is to the circumference as 7 to22, and the circumference to the diameteras 22 to 7. This is shown in the circleABCD, which, reduced to a straightline, as A F, is three times the diameterAC, and > of FE.

The second approximation is that thearea of a circle is equal to a right-angledtriangle formed of the circumference andradius of a circle; that is to say, theright-angled triangle GIH contains thesame area as the circle K, because thetriangle has its side H I equal to the cir-cumference of a circle, and G II, whichforms a right angle with it, equal to theradius G K. The other supposition ofthis great mathematician is, that thesuperficies of a circle is to the square ofits diameter as 11 to 14; that is to say,if the circle M contains 11 superficial feet,the square HP ON drawn on its dia-meter will contain 14, and it will beseen that the circle Q, inscribed in thesquare RSVT, is less than the square bya certain quantity.

Fig. 1141.

Fig.1143.

Fig. 1142.

Fig. 1144.

The relation of 7 to 12, which Archimedes gives as the ratio of the diameter of a circleto the circumference, holds good with a greater number of figures, as of 100 to 314, andthe larger the number of figures the nearer is the truth approached.

To find the Circumference of a Circle when the Diameter is given. By proportion, place inthe first term 7, in the second 22, and in the third the length of the diameter : the quotientwill give the circumference. If we require the circumference of the circle ABCD,according to Archimedes , we should place for a first term 7, for the second 22, and for athird the length of its diameter; the quotient would be the circumference sought: if thediameter be 15 feet for instance, then, 7 : 22 :: 15 : 47ft. 1 in. 8p.; or, 100 : 314 : 15 :47 ft. 1 in. 2p.

To find the Diameter of a Circle when the Circumference is given. Place in the first term22, in the second 7, and in the third the known circumference, and the quotient will be thediameter.

To find the Area of a Circle when the Diameter is given. First find the circumference, and