Chap. VIII.

GEOMETRY.

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Of Tables of Sines, Tangents and Secants and of Logarithms — In practical geometrythe instruments mostly used have their degrees usually divided into six equal parts, and con-sequently the tables are calculated to answer them by having their degrees divided into sixequal parts of 10 minutes each.

The sines, tangents, &c., are comprised in two sets of tables; the first contains the sumsof progression of common sines and tangents for the 90 degrees of the quadrant, for the100,000 parts into which its radius or entire sine is supposed to be divided. The secondtable contains the logarithms from 1 to 10,000 : in the first table the first column containsthe minutes of degrees, the second the sines answering in order to the minutes, the third andfourth the tangents and secants, the fifth the degrees, the sixth and seventh the logarithmicsines and tangents. The first column of minutes contains six divisions; the first is theminute, and then follows 10, 20, 30, 40, 50, CO, which answer to the five columns againstthem. The second column contains the sums of sines according to the progression ofdegrees and minutes of the radius or total sine, and the other columns contain tangents,secants, sines, logarithmic sines, and tangents ; whilst the fifth serves to mark the degreesfrom top to bottom to 45 degrees for the left-hand columns, and from bottom to top from45 to 90 degrees for those on the right-hand.

To find the Sine of an Arc, or the degrees of a given anyle and its complement —Every arc ina given angle may have less or more than 90°, and its degrees may have minutes or not.Required the sine of an arc of 24°, or simply the sine of 24°, look iti the table under thehead of degrees for this number, and against it is found 40673 for its sine ; but if it is re-quired to find the sine of 24° 10', we must further seek for 10' under the head of minutes,where, opposite to 24°, also in the column of sines, is 40939, which will be sine of 24° 10'.If it is required to seek the sine of 60° in the .table which goes from 45 to 90; when thesine is 140°, or exceeds a quadrant, or 90°, we must take the complement by subtracting140° from 180°, the value of a demicircle, and the remainder 40° must be sought in thecolumn of sines, which will give 64278.

To find the Degrees and Minutes of a given Sine , as that of the sine 40673.— Seek the sine inthe column of sines, and the figure opposite in the column of degrees will give 24^ andopposite this sine 40673 in the column of minutes is an 0, which signifies that the sine hasno minutes ; if the given sine had been 40939, seek it in the column of sines, and havingfound it in the left-hand columns, observe what degrees and minutes correspond with it,and having remarked it to be 24° 10', set it down as the sine required. The rules abovegiven for finding sines apply also to tangents and secants, both simple and logarithmic :when the distances measured are in feet or inches, it is necessary to reduce these dimen-sions into seconds, because these small fractions become less considerable, as fractions oflines in great distances are of little importance in a practical point of view.

To distinguish when observing objects, if the Sides of the Triangles formed are either Sines,Tangents , or Secants, — In a right-angled triangle, when we know the hypotenuse, or theside opposite the right angle, the sides of this triangle are considered the sines, because, iffrom one extremity of the hypotenuse, with its length as a radius, we describe a circle,it will inclose the whole triangle, and there will be neither tangent nor secant; supposing itrequired to find the distance from A to C, by employing a u

right-angled triangle ACB, of which the two angles 13 A Cand ABC are known with the hypotenuse B C ; it is evidentthat if from the point B, the extremity of the hypotenuseB C, and with the radius or length B C, we describe a circle,as DCEF, it will enclose the triangle ACB, and its sidesA C and A B will be sines, because they are enclosed in thecircle DCEF. Observe also, that in the same right-angled tri-angle A C B, that it is a general rule when one of the two sidesare known which form a right angle, as for example B A, we canuse it as a radius or total sine, to describe a circle, as A G III ;that the other side A C of the right angle B A C is a tangent,because it falls perpendicularly on the extremity of the radius13 A at the point A, where it touches the circle ; and that theside B C, which is opposite to the right angle BAC , is a secant, because it cuts the circleA G H I in K.

In trigonometry, all the sides of acute and obtuse-angled triangles are called sines,because these triangles, having no right angle, can have neithertangent nor secant of angles : if it be required to find thedistance LM by the obtuse-angled triangle LMN, it is clearthat having no right angle, it can be neither the tangent norsecant of an angle.

To measure lengths by the calculations of Sines, forming aright-angled triangle, of which the hypotenuse and two angles

arc known, viz. right and acute —To find the unknown angle, subtract the acute angle from

Fig. 1079.