Cmai*. Nil I.
GEOMETRY.
815
(lifters from an arc of that number of seconds only the fraction of a second. In planegeometry we consider angles as belonging to triangles which do not exceed 180 degrees,but we may fancy them of unlimited increase or diminution : if a line, for instance, revolveround a central point, it will in a revolution move through 3G0 degrees, and in a revolutionand a quarter, that number with the addition of 90. If we call 180 degrees tt, the revolvingradius in every revolution will move through the angle 2 tt, and in every quarter of a re-volution and in every half revolution through tt. Tn general, if n be an integer, the radius
after a number of complete revolutions will have moved through an angle expressed by2 n tt. If it has exceeded a complete number of revolutions by an angle w, the anglewhich it has described will be expressed by 2 n tt + w, and if it fall short of a completenumber, it will be expressed by 2 n tt — a*. If the angle it has described exceed an exactnumber of revolutions by half a revolution, we shall get its expression by changing winto tt in the former formula, which gives 2 n tt + it = (2 n + 1) tt. If, in like manner, theangle which the revolving radius has moved through exceed or fall short of a complete
number of revolutions by a right angle, its expression will be found by changing w into ^in each of the formula, which gives
2 n tt + ^ =(2 n + £) it, and 2 n tt — — =(2 n — £) tt.
Fig. 989.
The angle ~ — w is called the complement of w , and the angle tt — <a the supplement of w.
To find the length of the inclined line A 11, fix two piquets, one at C and the otherat I), and measure the anglesDC A and 1)CB, the firstbeing 27°, the other 42°.
Then from the piquet I),measure the angles CDB,
120°, and C D A, 142° : thenmeasure the distance be-tween the piquet C I), whichis 9 yards: construct thescale E, and set oft’ on theline FG nine parts takenfrom the scales, and then con-struct the two angles I F Hand KGL, and the distance from their points of intersection will be the length required.
Heights that are in-clined and inaccessiblemay also be measured,as that of the Leaning Tower at Pisa . Fromthe stations R and S,from whence may beseen the base and sum-mit, plant two piquets :then place the demi-circle at the piquet R,and measure the anglesS It O and S It P :place the demicircle atthe piquet S, and mea-sure the angles ItSP
and RSO, and measure the distances betweenthe piquets R, S: lay this down upon paper, andfrom the points where these angles unite or cut,as at c and rf, measure the length cd by the scale,and it will give the inclination of the tower, orrather its inclined height.
Depths which are inaccessible can also be as-certained, as that of the well A : measure thediameter A B, and at the point B measure theangle ABC with the demicircle and by a scaleof parts; the perpendicular A C, or depth, maybe ascertained by drawing the right angle C F I,and setting out the angle HLK, and from thescale taking the height L F.
Fig. 990.
Fig. 991.