LENGTH OF LINES.
The circumference of a circle is equal to the diameter multiplied by 3.14159265358979323846264338328 . . . . . This constant is usually denoted by the Greek letter m (pi) and is approximately equal to 3+, or 3.1416.
Use the first if an error of one two-thou sandth of the final result may be neglected, and the last if an error of one four-hundred-thoo sandth is negligible.
The above theorem may be written 1= ird, =2mr, where 1==.1ength of circumference, d =diam eter and r =radius.
The length 1 of the arc AMB in Fig. 1, in which MD is the perpendicular at the middle point of the chord AB, AE is tangent to the circle at A or perpendicular to the radius OA. OE is perpendicular to AM, the chord of lull the arc or the angle AOE is one-fourth of the angle AOB, may be found by any one of the formulas IT I 1 180 57 (error about lia) iII 180 8 hi 1---=c (III' 1 --= 2k+ 1(2k c), (IV1 j(8k c), 1=-- t st IV) in which d = number of degrees in the angle AOB, r=radius OA, c= chord AB, h =--- ei gh t MD, k= chord of half the arc AM, t =length of AE, and s of BE. Formulas (I11). (IV) and (V) are only approximate, the error being small only if the angle AOB or the ratio of h to c be small. Thus for an angle of dam'
the arc is 1.5708 . . . , whereas formulas an). (IV) and (V) give 1.576 .. , 1.5696 .. , . . , respectively.
Parabola.The length I of the arc AVB of the parabola in Fig. 2 is I= +4x2) where x° VD, y AD, and the logarithm is taken to the base e.
The length I of the circumfer ence ABA'B' of ellipse in the next figure (Fie.. 3), in which F and F' are the foci (I3F= BF' °OA), is 2za( I to 444 ... ), (VII) or approximately 1=IrV2(a2-F b7) (VIII) =4.443d; where e is the eccentricity (OF/OA a a".. OA, one-half the longest diameter; b = OB. one-half the shortest diameter ; use of formula (IV) is indicated by the diagram herewith (Fig. 4) and formula (IX). The only restrictions upon the location of the points 1, 2, 3, 4, .. . in addition to that given above is that the odd-numbered ones must be half-way between the others and that there should be an odd number of points in all.
1=(A140+23+ )+CA1+0+273+ .)