n-;
yif
where the numbers found answer to segments of differentproportions. The results found, both by Huygens ’ rule andby that of Nicholson, are compared with the result found byan infinite series, which is the criterion ; because, by suchan infinite series the answer may be found, which can bedepended upon, to any number of figures. The result foundby the series in the table is therefore true to the last figure,or to the last but one when the next figure would be above5. In this case, the last figure of the decimal is augmentedby unity.
In the table, the first vertical column contains the chords,and is marked c ; the second column contains the versedsines of the several segments, and is marked v; the thirdcolumn contains the lengths of the arcs, according to thedimensions stated in columns c and v in the same horizontalrow, and is marked x.; and the last column, marked K, showsthe ratio, or number of times that the versed sine is containedin the chord. In each cell under l are the three results ;the upper one being that of Huygens , the middle one that ofthe series, and the lower one that of Nicholson.
c
V
L
R
936
155
1002.666
1002.659
1002.9
24
5
26.666
26.687
26.69
41
556
136
631.333
632.252
632.5
S
3
10.666
10.724
10.743
2f
240
119
370.666
374.960
375.24
2*
14
7
21.73
21.991
22.
2
From this table it appears, that in low segments the reis nearly the same by each rule, but that Nicholson ^nearer to the series than that of Huygens ; and it ^ >eC< ^j J eSmuch more so in proportion as the segment app roa . gtowards a semicircle : thus, in a semicircle, the length o*arc found by the series is 21.991; the length found ^Huygens is 21.73 ; the difference .261. Now, the lengthe arc found by Nicholson’s rule is 22, which is ultin 111 j 9the Archimedean proportion, and the difference betweenresult and that found by the series is only .009. t j,g
In the first horizontal row, both methods agree wit* 1 - 0series to four places of figures ; in the second, they ag re6 j 0three places; in the third, fourth, and fifth, they ag re ® jtwo places ; and in the sixth they agree in one. In thehorizontal column, Nicholson’s method agrees in five P 1 . 8 e8with the series ; in the second, third, and fourth, with *figures of the series; in the fifth, with two figures ot yseries, and in the last with one figure of the series ; hutdiffers in five figures from the truth by .009. , .- n o
The length of the arc found by the two approxim® ,rules in low segments, is exceedingly near the true va ge g-but when the result is found by Huygens ’ rule in highments, it is very considerably below it, and cannot we*employed in large dimensions.
There is still another rule to be found in books of 1116 ^ration, which will give the length of the arc to any exact 1 * ^either by finding the number of degrees by means 0 ^instrument, or by trigonometry; the former is n °^ uC fieligible in practice, and the latter would require too a*time to be of real utility amid the hurry of business. e0 -The investigation of Eule 6, Theorem X., from the Sym-metrical division of the arc of the circle into equal V ^by the diameter being divided also into equal parts,follows : j 0 ter'
Let A d b c be a circle ; A b and c d two diameters, er ,secting each other in e at right angles ; let the dia . rC jeD C, be produced to f, so that the part, c f, without the 0 fmay be three-quarters of the radius, e c, or three-e'g**the diameter, d c ; draw d g parallel to A»b ; tax j, jportion, d i, of the arc D A ; join F I; and prod uto g :
Let DC be denoted by d; then will fd = +
8 8
and F K ” FD — K D =
11c?
■V =
11c? — 8 v
8 “ 8
Now, the triangles f k i and f p g are similar; therefore f k ; fd ; : kt ; dg;lid — 8® 11 d lids
that is,
8
8
DG ==
lid —8®
; which is the value of the tangent.
Then, to show how nearly this value is to that of the arc, we have
11 ds
lid — 8v 1 —
but
s =(d»—»*)£ = d£ a2 c i _ i —
X 2d 8d>’ /
8 v .Ud’
therefore
2d
V v 1
[ _ i x 1 ~ YT~ TdP &c -
8 v = v 2 X ^7, -
= x(i + |^ +
lid
I —
39 v 1
22 d 968 d a
■)
the
Jo*
l t