Chap. VIII.

GEOMETRY.

861

Fig. 1090.

angle opposite to the side to be known, to add to the lo-garithm already figured a part, and subtract their sum fromthe logarithm of the sine lOOOOOOOO, which is the end of thecolumn of logarithmic sines, where the 89° 60 of the tablesof sines end ; the remainder of this subtraction will be alogarithm, which sought in the table of logarithms will givethe value of the required side.

Let it be required to find the distance A B: having formedthe right-angled triangle ABC, of which the hypotenuseC B is known to be 1296 inches, and the acute angle ACB50°, with the right angle CAB 90°. To find the third angleC B A, subtract the acute angle ACB 50° from 90° ; theremainder, 40°, will be the acute angle CB A. To find theinaccessible side A B, seek in the column of numbers the sum of the hypotenuse B C,1296, which will give the logarithm 31126050. Then seek in the column of sines oflogarithms the logarithmic sine of the angle A C B, 50°, opposite the side required, A B ;the logarithmic sine will be 98842540, which add to the logarithm figured aside, andfrom their sum, 129968590, subtract the total logarithmic sine 100000000; the remainder,29968590, will be a logarithm, which being sought, or its nearest number, 29969492, willgive opposite in the column of numbers 993 for the length of the side A B in inches :according to the same rule it will be found that the side A C is 833 inches long.

Measuring Lengths, by forming a right-angled triangle, of which one of the sides whichform the right angle is known, with the adjacent right and acute angles. To have thethird unknown angle subtract the known acute angle from 90°; to have the other side,forming the right angle, which is a tangent, seek in the column of numbers the length ofthe known side, and write down the logarithm opposite it.

Then seek in the table of sines the tangential logarithm ofthe angle opposite the side to be found, which add to thelogarithm written down, and from their sum subtract thetotal logarithmic sine 100000000; the remainder will be alogarithm, which being sought in the column of logarithms,will give in the opposite column of numbers the side re-quired.

Required the height A C of the right-angled triangleA B C, where the side A B is 993 inches, the right angleCAB 90°, and the acute angle C B A 40°. We know thatby subtracting the acute angle C B A, 40°, from 90°, theremainder 50° will be the unknown angle A C B. To havethe height A C , which is a tangent, seek in the column of numbers the side A B, 993inches, to take its logarithm 29969492 : seek in the table of sines the tangential logarithmof the angle C B A 40°, opposite the side required ; the tangential logarithm will be99228135, which add to the logarithm before found; their sum will be 129207627 ;subtract the total logarithmic sine 100000000 from it; the remainder, 2920762, will bethe logarithm, which sought in the table of sines will give 833 inches, opposite its nearestnumber, 29205430, for the height required of AC. * To have also the hypotenuse C B,add the total logarithmic sine, 100000000, to the logarithm 29206450, and from theirsum, 12906450, subtract the logarithmic sine 98080675, of the angle CBA 40°; theremainder, 31125775, is a logarithm, which sought in its nearest number, 31126050, inthe table, will give 1296 for the hypotenuse C B.

Of measuring Lengths by forming a right-angled Triangle , of which the hypotenuse andone of the sides which form the right angle and this angle are known. To find the thirdunknown side, add to the length of the hypotenuse thatof the known side; seek their sum in the column of numbersand figure the logarithm, which is opposite, by itself; thensubtract the known side from the value of the hypotenuse ;seek the remainder in the column of numbers, take the lo-garithm opposite, to figure it above or below the last loga-rithm ; add the two together, and take half their sum, whichbeing a logarithm, seek it or the nearest number to it inthe table, the number opposite marks the length of theunknown side.

Required the distance from A to B having formed aright-angled triangle ABC, of which the hypotenuseC B is 1296 inches, and the side AC 833 inches, with theright angle CAB, 90°.

To find the side A B, add the hypotenuse 1296 to the side*A C 833 ; seek their sum inthe column of numbers, which will give the logarithm 33281757. Then subtract the side

Fig. 1091.

Fig. 1092.