591 1-71286

•592 1-71546

593 1.71707

||

|

634 1.78335 || ·677 | 1·85379 || ·720 | 1·92531
|
635 1.78498 || ·678 | 1·85544|| ·721 1.92700
•6361-78660 || ·679 | 1·85709 722 | 1.92868
637 1·78823 || ·680 | 1·85874 723 1.93036
|
•638 1.78986 •6811.86939 -724 1.93204
682 1.86205 725 1.93373
683 1.86370726 1.93541
·6841.86535727 1.93710

•594 1.71868
•5951.72029
596 1.72190 639 1.79149
.597 1.72350 640 1.79312
641 1.79475

598 1.72511

599 1.72672 •642 1.79638 685 1.86700 || ·728 | 1.93878 600 1.72833 643 1.79801686 1.86866 || 729 | 1·94046 | •601 1-72994 644 1.79964|| 6871-87031

•602 1-73155 645

603 1.73316 646

730 | 1.94215

1.80127 || ·688 1.87196 7311.94383 1.80290689 | 1·87362 || 732 1.94552 .604 1.73477 .647 1.80454 || ·690 | 1·87527 || ·733 | 1.94721 ·605 1.73638 648 1·80617 |

•6911.87693 || ·734 | 1·94890 •606 1.73799 •649 1.80780 692 1.87859 || ·7351.95059 ·6071-73960 || ·650 | 1·80943|| 693 1.88024736 1.95228 .694 1.88190 737 1.95397 ·695 1.88356 || 738 | 1-95566 ·6951-88356||

.614 1.75091 657 1.82091700 1·89186 |

| || ·743 743 1.96414 .6151-75252 || ·658 | 1·82255 701 | 1-89352 ·744 | 1.96583 -6161-75414 659 1.82419702 | 1·89519 || ·745 | 1·96753

·703 | 1·89685|| ·746 | 1.96923 618 1-75738 || ·661 | 1·82747 .704 1.89851 ·747 1.97093 619 1.75900 || ·662 | 1.82911 705 1.90017 •748 •620 1.76062 || ·663 | 1·83075706 | 1.90184 749 •621 1.76224 ·664 1.83240 || ·707 | 1·90350 750 1.97602 •622 1.76386 665 1.83404 708 1-90517 -751 | 1.97772

·626 1·77034 | 669 1-84061 || ·712 | 1.91187 755 1-98453

758 1.98964 801 2.06377 .844 2.13976 887 2.21754

·759 1.99134 || ·802 | 2.06552 || ·845

888 2.21937

·802 760 1.99305 || ·803 .803 ·761 | 1·99476 || ·804 ·7621-99647 || ·805 763 1.99818 .806 •764 1.99989 || ·807 765 2.00160

2.06552|| ·845

2.06727 846

2.14155
2.14334

2·06901

889 2.22120 .847 2.14513|| 8902-22303

.805

7662.00331

-855 2.15950 770 2.01016 813 2.08480 856 2.16130 771 2-01187

.809 2.07777852 2.15409

·767 2.00502 810 2.07953 853 2-15589 768 2.00673 811 2.08128 854 2.15770 897 2.23590 | 7692-00844 812 2.08304

2·07076 2.07076 848 2.14692 .891 2.22486 2.07251 849 2.14871 -892 2.22670 807 2.07427 .850 2.15050 .8082.07602 851 2.15229 || |

-893 2-22854

.894 2.23038

.8952-23222

.814 2.08656 ·857 2.16309

.900 2-24142

774 2.01702 || ·8172·09198 ·8602-16848 903 2.24691

775 2.01874 818 2.09360
776 2.02045|| ·8192-09536
777 2.02217 ·820 2.09712
778 2.02389 821 2.09888
779 2.02561 822 2.10065
·780 2-02733 823 2.10242
781 2.02907 824 2.10419 .867 2.18113
-782 2.03080 825 2.10596|| 868 2.18294
-783 2.03252 826 2-10773 869 2.18475
·784 2.03425 827 2.10950 8702.18656
·785 2.03598 828 2.11127 .871 2.18837
786 2.03771 829 2.11304 872 2.19018
787 2.03944 8302-11481 873 2.19200
·8312.11659 .874 2.19382

Height

Length Height Length Height Length of Arc. of Arc. of Arc. of Arc. of Arc. of Arc.

.928 2.29270 947 2.32785 || 9652-36191
|| |
|

.983 2.39631

·966 2.36381

•984 2.39823

.967 2.36571

•985 2.40016

.968 2.36762

.986 2.40208

•929 2.29453 | 948 2-32972

•930 2.29636·949 2.33160
•931 | 2.29820 || ·950 | 2·33348
•932 2.30004 951 2.33537 •969 2.36952
933 2.30188 952 2.33726 || ·970 | 2·37143
|| |
934 2.30373 || 9532-33915 || ·971 | 2·37334

987 2.40400

•988 2.40592 -989 2.40784 .990 2-40976 ·991 2-41169 -9922.41362

•935 2.30557 954 2.34104|| 972 2.37525 | || | •936 2.30741 || ·955 2.34293 973 2.37716 -937 2.30926 956 2.34483 974 2.37908 | 938 2.31111 957 2-34673 || ·975 2.38100 -9392-31295 || ·958 2·34862 || ·976 | 2.38291 .940 2.31479 959 2·35051 .977 2.38482 .941 2.31666 ·960 2.35241 .978 2.38673 -942 2.31852 961 2.35431 .979 2.38864 •943 2.32038 || ·962 | 2.35621 .980 2.39055 .944 2.32224 963 2.35810 | 945 2-32411 ·964 2·36000 || ·982 | 2·39439|1.000 2.42908 || | .946 2-32598

.993 2.41556

.994 2.41749

.995 2.41943

•996 2.42136

-997 2.42329 .9982-42522

.981 2.39247 •999 2.42715

|

To find the Length of an Arc of a Circle, or the Curve of a Right Semi-Ellipse.

RULE.-Divide the height by the base, and the quotient will be the height of an arc of which the base is unity. Seek, in the Table of Circular or Semi-elliptical arcs, as the case may be, for a number corresponding to this quotient, and take the length of the arc from the next right-hand column. Multiply the number thus taken out by the base of the arc, and the product will be the length of the arc or curve required.

EXAMPLE 1.—In Southwark Bridge, London, the profiles of the arches are the arcs of circles; the

span of the middle arch is 240 feet, and the height 24 feet; required the length of the arc.

24240100; and 100, as per Table V., is

1.02645.

=

Hence 1.02645 x 240 246-34800 feet, the length required.

EXAMPLE 2.-The profiles of the arches of Waterloo Bridge are all equal and similar semi-ellipses; the span of each is 120 feet, and the rise 28 feet; required the length of the curve.

28120233; and 233, as per Table VI., is

1.19040.

Hence 1.19010 x 120 = 142-81200 feet, the length required.

In this example there is, in the division of 28 by 120, a remainder of 40, or one-third part of the divisor; consequently the answer, 142.81200, is rather less than the truth. But this difference, in even so large an arch, is little more than half an inch; therefore, except where extreme accuracy is required, it is not worth computing.

These Tables are equally useful in estimating works which may be carried into practice, and the quantity of work to be executed from drawings to a scale.

As the Tables do not afford the means of finding the lengths of the curves of elliptic arcs which are

less than half of the entire figure, the following geometrical method is given to supply the defect.

Let the curve, of which the length is required to be found, be A B C.

Produce the height line, B d, to meet the centre of the curve, in g. Draw the right line, A g, and from the centre, g, with the distance, g B, describe an arc, Bh, meeting Ag in h. Bisect A h in i, and from the centre, g, with the radius, g i, describe the arc ik, meeting d B, produced to k; then ik is half the arc A B C.

6*