Chap. VIII.
GEOMETRY.
821
rectangle under the two diagonals is equal to the sum of the two rectangles under theopposite sides. This system of computation by chords was changed for that of sines bythe Arabians, who improved the science left by the Greek school.
The division of the line of solids is drawn below that of the chords, and of the same length.As the line of planes is founded on the knowledge of the equal sides of perfect squares,that of solids is founded on the cube roots of the equal sides of cubes, which are alsomarked up to 64. The first point of the line of solids is placed opposite the division 50of the line of equal parts, and is as follows : —
1
opposite
50
17
opposite
128$
33
opposite
160$
49
opposite
1S3$
2
—
63
18
—
134ft
34
—
162$
50
_
184$
3
—
72
19
—
133$
35
—
163$
51
_
185^
4
—
79$
20
—
I35J
36
—
165$
52
_
186$
5
—
85$
21
—
138
37
—
166$
53
_
187$
6
_
91
22
—
140
38
—
168$
54
_
189
7
_
95|
23
—
142
39
—
1691
55
_
190$
8
_
100
24
—
144
40
—
171
56
_
191$
9
—
108ft
25
—
146
41
—
172$
57
—
192|
10
—
111$
26
—
148
42
—
174
58
—
193$
11
—
114$
27
—
150
43
—
175$
59
—
194|
12
—
1161
28
—
151$
44
—
176$
60
—
195
13
—
117$
29
—
153$
45
—
178
61
—
196$
14
—
120$
30
—
155$
46
—
179$
62
—
197$
15
_
123*
31
—
157
47
—
180$
63
_
198|
16
—
126
32
—
158$
48
—
182
64
—
200
The division of the line of metals is divided according to the differences in the calibre ofthe balls, and each metal is distinguished by its particular sign : that of gold the sign isput opposite 146$ of the line of equal parts, that of lead 172.*, that of silver 179, that ofcopper 1873, that of iron 195, and that of tin 200.
To divide a right line into two or more equal parts by the sector, , ..
take its length with a pair of compasses, and carry it to the line of a b
equal parts, then keeping the sector open, carry from the line ofequal parts the number which exactly divides it. Supposing it is
required to divide the line into five parts, which shall be equal, take its length and set itoff on the line of equal parts on the sector at the opening of a number divisible into asmany parts as the line is to be divided, viz. it must be remarked, first, what number isdivisible by 5, and having observed that 200 is one of those numbers, as 40 is the fifth ;place, therefore, the two points of the compasses containing thelength of the line to be divided on each point of the 200, which A B
is usually at the extremity of the sector, and keeping it open, - -i
then take the distance between the two 40 on the line of equalparts, which distance will be only one-fifth of the length of the
Fig. 1009.
line.
To divide a circle by applying a right line as often as is re-quired : divide 360 by the number of times it is required to ap-ply the given line, and the quotient thus obtained is so manydegrees. Then open the sector and carry the length of the givenline to the line of chords, where the figures answer to the numberof degrees previously obtained : keep the sector open, and take theopening between the two 60° on the same line ofehords: this openingwill be a radius to describe a circle, which may be divided by thegiven line into as many parts as is required, or into five, as shownat CDEFH.
To divide a circle into equal parts by the sector, take in yourcompasses the length of the radius, and carry it to the two 60° onthe line of chords; then divide 360 by the number, as alreadydescribed. For example, it is required to divide the circleA B C D E into five equal parts : take the length of its radiusF B, and carry it to the two 60° on the line of chords, opening andshutting the sector, until the two points of the compasses, open tothe extent FB, fall exactly on the two 60° of the line of chords.Keeping the sector at this opening, divide 360 by 5, the numberof equal parts required: we shall have 72 as the quotient, whichmust be taken from the line of chords as before ; this opening willthen precisely divide the circle into five equal parts.
3 g 3
c
Fig.1010.