820
THEORY AND PRACTICE OF ENGINEERING.
Book II.
line of plans; is placed opposite the 25th division on the line of equal parts, and the secondpoint opposite 35 and continued as in the following table.
1
opposite
25
17
opposite
103
33 opposite
i «*
49
opposite
175
2
—
35^
18
_
106
34 —
146
50
—
176$
3
—
434
19
_
109
35 —
148
51
_
178$
4
—
50
20
_
ui$
36 —
150
52
—
180$
5
—
561
21
_
114$
37 —
152
53
—
182
6
—
61 Vs
22
_
H'$
38 —
154
54
—
183$
7
—
66*
23
—
H9*
39 —
156
55
—
185J
8
—
70j
24
—
122$
40 —
158
56
—
187$
9
—
75
25
_
125
41 —
160
57
_
189
10
—
79
26
_
127$
42 —
162
58
—
1903
11
—
82*
27
—
130
43 —
164
59
—
192$
12
—
86$
28
_
132$
44 —
166
60
_
193$
13
—
90
29
_
135$
45 —
167$
61
—
195$
14
—
95$
30
_
137
46 —
1694
62
—
197
15
—
96*
31
—
139$
47 —
171$
63
—
198$
16
100
32
—
141$
48 —
173$
64
—
200
The lines of polygons are constructed by the division of circles,
or
by the proportion it
bears to the line of equal parts.
12
is opposite
60
9
is opposite
80
6 is opposite
116
4
is opposite
163
1 1
_
65
8
—
88
5 —
136
3
—
200
10
—
72
7
—
101
The division of the line of chords or angles is so named from its forming all kinds ofangles, either on paper or on the ground ; it is generally the same length as that of theequal parts, and is always di-vided into 180°, the number ofdegrees which a demicircle con-tains.
The line of chords is setout by describing from its middle,
K, as a centre, with the radiusK H, the semicircle IIL C,which must be divided into 180parts or degrees, so that the lineof chords shall be the diameterof the demicircle II L C. Fromthe point II, the centre of thesectors, place one foot of a pairof compasses, opening themto the first point of the divisionof the demicircle, and describean arc from thence cutting theline of chords at the first pointof its division : then from the same centre II, describe an arc from all the other divisionsof the demicircle cutting the line of chords II C : it will then be found that this line willbe divided into 180°, commencing their enumeration from the centre of the sector at II.
The ancients worked their trigonometry by means of chords and arcs, which, w'ith thechords of their supplemental arcs and the constant diameter, formed all kinds of right-angled triangles. Beginning with the radius, and the arc whose chord is equal to theradius, they divided them both into sixty equal parts, and estimated all other arcs andchords by those parts; viz. all arcs by 60ths of that arc, and all chords by (SOths of itschord or of the radius: this method is as ancient as the writings of Ptolemy , who usedthe sexagenary arithmetic for this division of chords and arcs.
Menelaus , at the commencement of the Christian a’ra, wrote six books on the chords ofarcs, and his system of trigonometry was greatly improved in the following century byClaudius Ptolemaeus , who taught astronomy at Alexandria: in the first book of hisAlmagest he has a table of arcs and chords, with their method of construction; it containsthree columns ; in the first are the arcs for every half degree, in the second the chords,expressed in degrees, minutes, and seconds, of which degrees the radius contains 60, andin the third column are the differences of the chords, answering to one minute of the arcs,or the thirtieth part of the differences between the chords in the second column. In thistable we discover the property of any quadrilateral inscribed within a circle, viz. that the
Fig. 1007.