Chat. VIII.
GEOMETRY.
863
be written below the sum total 130201982, and subtracted from it; the remainder, 90857*145,will be a tangential logarithm, the nearest number to which, 96849681, will give 25° 50 7 ,which being added to the 55° 30', the approximating value of the two angles, will give 81°2<y for the greater angle CBA, and 25° 50 / , being subtracted from 55° 30' there will remain29° 40' for the angle C A 13. To find the third side A 13 take the logarithmic sine of theangle A C 13, 69°, opposite the side A 13 ; this sine will be 99701517. Then seek in thecolumn of numbers the logarithm of the side BC, 720, which will be 28573325 ; figure itbelow the logarithmic sine 99701517, and add them together; the sum will be 128274842.Seek the logarithmic sine of the angle C A B, 29° 40', opposite the known side 13 C ; it willbe 96945642, winch subtract from the sum 128274842, the remainder, 31329200, will be alogarithm, the nearest number to which in the table will give 1358 for the distance in feetfrom A to B.
Measuring Lengths by forming an obtuse-angled Triangle , of which two angles and the adjacentside are known : the rules for the first acute-angled triangle must be applied here ; never-theless, since in the table of sines there are no logarithmicsines for obtuse angles, we must subtract the obtuse anglefrom 180°, and take the logarithmic sine for the remainderor complement. Required the distance A 13, havingformed the obtuse-angled triangle, A B C, of which theangle C A B is 33°, the angle A C B 109°, and the adja-cent side AC 312 feet. To find the third angle CBA,add the two angles C A B, 33°, and A C B, 109°, together,and subtract their sum 142° from 180°, the remainder 38° will be the required angle C’ B A.To have the length of the required side A 13, seek the logarithmic sine of the angle A C B.109°, opposite the required side A 13; but since the angle A C 13 109° is obtuse, subtract109° from 180°, the remainder 71° is its complement; the logarithmic sine of which is99756701. Then seek in the column of numbers the side AC 312; write down the loga-rithm which is opposite 24941546, below’the logarithmic sine 99756701, and add themtogether; their sum will be 124698247. Seek the logarithmic sine of the angle CBA 38°opposite the side AC, it will be 97893420, which subtract from the sum 124698247, theremainder 26804827 will be a logarithm, the nearest number to which in the table, 26803355,will give 479 for the distance A 13 in feet.
By the same rule it will be found, that the side C B is 276 feet, remarking that there isno sine of the complement to be taken for the angle C A B because it does not exceed 90°.
As for the other obtuse triangles, they are solved like acute-angled, only taking the com-plements of the obtuse angles.
Mensuration or Superficies and Surveving. — Of Bevelsand llecipiangles. — A bevel may be compared to a pair oflarge compasses, and is usually made of iron or wood : theiron bevel A, used by stone masons, has each of its branchesabout 2 feet long, smooth and pointed, and its head, w hich isround, will open to any angle required.
The carpenter’s bevel, B, made of wood, is commonlyshorter than the stone-mason’s; the extremity which forms itshead is cut at right angles, so that opening them to the squaremark they may serve more easily to express a right angle.
The Recipiangle C consists of two long rules of wood,which have their brandies parallel at their edges, and areattached together at the middle of their ends by a double-headed screw, which forms the head of the instrument.
When used for taking salient and re-entering angles, thecentre of a protractor of wood or horn, 6 or 8 inches indiameter, is applied at D to the point at which the branchescross, to observe how* much the branches are opened, andwhich give the required angle; in default of this instrumentwe use a bevel.
The recipiangle E, made of wood, copper, &c., is com-posed of two plates, each about a line in thickness,12 incheslong, and 3 inches wide : sometimes its length is increasedby adding slips of wood or flat rulers. At the extremity Fig. 1090.
of one of the two plates is described a demi-circle, equal to itsbreadth, divided into 180° ; and at the extremity of the secondblade, towards its centre, is a small tongue or round head,which is attached to the centre of its semicircle on theother limb by a double-headed screw.
The'instruments described are extremely simple and havethis fault, that it is difficult to apply the centre of a protractor
Fig. 1098.
A
Fig. 1096.
Tig. I ICO.