862

THEORY AND PRACTICE OF ENGINEERING.

Rook If.

A C , 833, from the hypotenuse C B, 1296 ; their remainder, 463, seek in the column of num-bers ; write the logarithm, 26655810, which is opposite, below the first logarithm ; add thesetwo together, their sum will be 59937567, and its half 29968783^ will be a logarithm ; seekthis or its nearest number 29969492 in the column of logarithms, which will give in thecolumn of numbers 993 for the number of inches from A to B.

Of measuring Lengths hg forming a right-angled Triangle of which the two sides forming theright angle and the latter are known. — Required the distance B C, having formed aright-angled triangle A B C, of which the side A B is 82 Pfeet, 8 inches, 9^ seconds, or rather more than 993 inches,the side AC 833 inches; and the angle BAC 90. Oneof the two acute angles must be found, say CB A, oppositeto the shortest known side A C , 833 inches. Write downthe total logarithmic sine 100000000. Then seek in thecolumn of numbers the figure 833, the value of the sideA C , which is that opposite the angle required, C B A ; andtake the logarithmic sine, and add them together : theirsum will lie 129206450. Seek in the column of numbers,the figures of the other side AB, 993 inches, and havingfound them, take the opposite logarithm, 29969492, whichwrite below the sum 129206450; subtract one from theother, the remainder, 99236958, will be a tangential logarithm, which, or the nearest num-ber to it, 99238135, you must seek in the column of tangential logarithms; in the oppositecolumn of degrees will be found 40° for the angle C B A: if you subtract this angle from90° there will remain 50° for the other acute angle A C B. Lastly, to find the distanceB C, which is a secant, add to the total logarithmic sine 100000000, the logarithm29969492 of the side A B, 993 inches, and from their sum 129969492, subtract the loga-rithmic sine of the angle AC B, 50°, which is 98842540, the remainder 31126952 is alogarithm; which, or its nearest number 31126050, will give 1296 for the required distanceB C, in inches.

Measuring Lengths by forming an acute-angled Triangle , of which two angles and the adjacentangle are known : to find the third angle, add together the value of the two known angles,and subtract their sum from 180° ; the remainder will be theunknown angle. To find the two unknown sides, as thatwhich is opposite to the least angle, seek the logarithmicsine of the angle, and write it down: then seek in the columnof numbers the value of the known side, figure its logarithmbelow the logarithmic sine before found, add the two toge-ther, and from their sum subtract the logarithmic sine ofthe angle opposite the known side; the remainder will be alogarithm ; which being sought in the column of logarithmswill give in that of numbers the length of the unknown side.

Required, in the acute-angled triangle ABC, the dis-tance from A to B: the angle A C B, being 79°, CBA,

70°, and the adjacent side C D, 3480 inches. To find theangle CAB, add together the two angles A C B 79° and CBA 37° : subtract their sum116° from 180°, the remainder 64° will be the unknown angle CBA. To have the length ofthe side A B, seek in the table the logarithmic sine of the angle A C B, 79°, opposite the sideA B required to be found ; this logarithmic sine will be 99919466 : then seek in the columnof numbers the length of the known side CB, 3480 inches, which will give opposite the lo-garithm 35415792, which write below the logarithmic sine 99919466; add them together,their sum will be 135335258. Lastly, seek in the table the logarithm of the angle CAB 64°,and it will be 99536602, which subtract from the sum 135335258, the remainder, 35798656,will be a logarithm, the nearest number to which, 35797836, will give 380 for the distancein inches from A to B.

Of measuring Lengths , by forming an acute-angled triangle, of which two sides and thecomprised angle are known. Required the distance from A to B of the acute-angled tri-angle A B C, of which the side A C is 1440 feet, C B 720,

and the angle ABC 69°. To find one of the two unknown B A

angles, subtract the short side B C, 720, from the long oneA C , 1440, their difference, 720, will give the logarithm,

28573325. Subtract the known angle A C B 69° from180°, the remainder 111° is the value of the two unknownangles, of which take the half, 55° SO', their tangential lo-garithm in the table of sines will be 101628657, which addto the logarithm 28573325; their sum will be 130201982. cThen add together the side^A C 1440, and B C 720 ; their Fig. 1095.sum, 2160, will give the logarithm 33344537, which must

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Fig. 1094.