Chap VII 1,

GEOMETRY.

853

would shorten to a certain degree the measurement, or the flexure of the rods wouldproduce the same effects. Two steel chains, each 100 feet in length, and containing fortylinks, made by Ramsden, were selected to remeasure the line: the links were inthe form of a parallelogram, of ^ inch square and 30 inches in length. When used,they were strained over five boxes placed at equal distances upon bricks, and stretched in astraight line by weights of 56 pounds : to bring the extremities of the two chainstogether over the same point, they were supported at each end by an upright piece ofwood or post; that of the preceding end had attached to it a pulley over which passedthe rope which stretched the chain, whilst to the end that followed was applied a screw ap-paratus, by which the chain could be drawn back against the weight: another upright orpost at each end, not connected with the others or with the chain, supported a scale.

The chain being placed, the scale at the preceding end was moved by means of screws,until one of its divisions exactly agreed with the mark on the handle of the chain : thescale remaining in its place, the chain was then carried forward, and again adjusted by thescrew scale in a similar manner : after 38 of these chains had been used, one in con-sequence of the links being bent was laid aside, and the base line was measured by theother ; the one laid aside being repaired, was retained as a standard. Experiments out ofnumber were made previously, to ascertain the rate of expansion, and a thermometer wasplaced at each of the five boxes when the measurement was taken, in order to insureaccuracy ; the chains, after the measurement, were again compared, when it was found thatthat which had been used had lengthened through the rubbing of the joints *0373 of an inch.

The length of the base line was by means of the chains found to be 27404*3155 feet,being about 2^ inches more than that measured with the glass rods. The mean of the tworesults, 27404*2 feet, was then assumed for all the future calculations. Rods seem, how-ever, in general, to have a preference over the chain, where great accuracy is required,as they cannot be considered so flexible, nor are they so likely to increase in length, asthe chains when slightly strained may do at the joints: great care must be taken atthe commencement of such an operation, in ascertaining the exact length of the foot, tobe marked on those measuring rods or chains, and some standard should be applied to forthe purpose. Such a standard is now adopted as constitutes the yard at 36*000659 inches,the length of the pendulum vibrating seconds being taken at 39*13929 inches; this iscalled the imperial measure, and as it was supposed to be a correct and unalterablequantity, it was prescribed by Act of Parliament that the length of the standard yardshould be restored by reference to the length of a pendulum, should accident occur toinjure the legal standard deposited in the House of Commons .

In Ireland another method was adopted to measure the base line set out on theplains of Magellan, which is between 7 and 8 miles in length, and where the greatestpossible error is not supposed to exceed 2 inches. Two bars, each 10 feet long, one ofbrass, the other of iron, were placed parallel to each other, and riveted at their centres, ithaving been previously ascertained by experiment that they expanded or contracted in theproportion of three to five; some nonconducting substance was spread over the brass bar,which equalised the two metals in their susceptibility as to change of temperature : atongue of iron, with a minute dot of platina, was fixed across each extremity of these twocombined bars, and this tongue had the dots so placed that under every change of con-traction or expansion they remained at the constant distance of 10 feet. The tongues wereperpendicular to the rods, at the temperature of 60 degrees of Fahrenheit, and the ex-pansion of the two bars of brass and iron taking place from their common centre, as theinclination of the tongues became changed, the platinum dots remained unalterably fixedat the exact distance of 10 feet. When the base line is accurately taken, it shouldbe reduced to its proper measure at the level of the sea, and as this is constantly changing,some elevation must be established that can be referred to, and that made use of by theOrdnance surveyors is low water mark.

Suppose R = to the radius of the earth at the level of the sea; R + A equal to the radius atthe level of the measured base; A = to the measured base A B, and a = to the reduced base a b.

Then, as similar arcs are in the same ratio as their radii, we have R + A ; R : : A : o, and

A =

R. A

R-j-A

and the difference between the measured and reduced base is

A —a = A —

R A

R + A *

reducing both terms to a common denominator :

. AR + AA-AR

A -“=-R ~h -

whence

A — a = ——- = difference required.

R + A 1

The elevation of Hounslow Heath is about 102 feet above the level of the sea, and the

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