a mechanical device: the sector compass alfredo rodriguez department of mathematical sciences...
TRANSCRIPT
A Mechanical Device:the sector compass
Alfredo Rodriguez
Department of Mathematical SciencesUniversity of Texas at El PasoOctober 16, 2001
Outline
Sector compass Description of device Different scales
Lines of lines Lines of superficies
Application Building your own sector
compass
p1’’’
10'
0
1 1'
2 2'
3 3'4 4'
5 5'6
6'7 7'
8 8'9 9'
10
Sector Compass
Introduction
Invented by Guidi Ubaldo de Monte The device has different scales
Lines of line Lines of superficies
The compass was used for about two centuries
Description of Sector Compass
Given that AB=AC. If AE=AD, then AC and AB are cut proportionally.
Draw in BC and DE
AB:AD=BC:DE AB and AC are legs of sectors
A
BC
DE
Sector Compass
10'9'
10
8'
9
7'
8
6'
7
5'
6
4'
5
3'
4
2'
3
1'
21
0
0
1 1'
22'
33'
44'
55'
66'
7
7'
8
8'
9
9'
10
10'
Scales of Line of Lines
Each ruler has equally space marks starting from point A.
A
BC
11
2
3
4
2
3
4
Application for the Sector Ruler using line of lines
p
q
p A
6 6
1 1
2
3
4
5
2
3
4
5
Increase a given line segment p by proportion of 3:5.
Given a segment, p. Open rulers up until p fits into 3-3, then find 5-5 segment.
This will be the desired length.
How tall is the chair?
This artwork is outside of Denver Public Library.
The horse is 7 ft tall.
Chair to Horse Proportion
0
12
1'
3
2'
4
3'
5
4'
6
5'
7
6'
8
7'
9
8'
10
9'10'
1’2’
1
3’
2
4’
3
5’
4
6’
5
7’
6
8’
7
9’
8
10’
910
1’2’
1
3’
2
4’
3
5’
4
6’
5
7’
6
8’
7
9’
8
10’
910
1’2’
1
3’
2
4’
3
5’
4
6’
5
7’
6
8’
7
9’
8
10’
910
10'9'
10
8'
9
7'
8
6'
7
5'
6
4'
5
3'
4
2'
3
1'
21
0
Chair to Horse
x =70
3 ft » 23.333ft
3
10=
7 ft
x
Lines of Superficies 0
1 1'
2 2'
3 3'
4 4'
55'
66'
77'
88'
99'
10 10'
The marks are not equally spaced.
Creating the Divisions (Superficies)
Point A: center of Circle
AB perpendicular to AC
Marks of AD are 1/10 of AD
AE= ½ AC
Marks of AE are 1/10 of AE
Create circles with center AE1 and radius AD1
Find intersection this circle and line AB and call it B1
This will be the first mark on the side of each ruler.
Continue with each point ADi , AEi and ABi.
Scales for Line of Superficies
DC
B10
E A9A8A7A6A5A4A3A2A1E9E3E2E1
B1
B2
B3
B4
B5
B6
B7
B8
A
B9
Find scale for lines of Superficies
A1A DA9A8A7A6A5A4A3A2A1AE9E3E2E1E AC AC E1
B1
A A1
B10
B7
B8
B9
DC E A A9A8A7A6A5A4A3A2A1E9E3E2E1
B1
B2
B3
B4
B5
B6
Values of ABi
i CA AAi Exact ABi ABi1 10 1 10 3.16228
2 10 2 25 4.47214
3 10 3 30 5.47723
4 10 4 210 6.32456
5 10 5 52 7.07107
6 10 6 215 7.74597
7 10 7 70 8.3666
8 10 8 45 8.94427
9 10 9 310 9.4868310 10 10 10 10.
Formula of length of ABi
LengthofABi givenby
b HiL=10* i =
10 i for0 £ i £ 10.
Compare the Area of two Squares
Given two square with side a and b. Use the line of superficies scale for this calculation.
p2
p1
a b
Compare AreasPlace side a on the 10-10 line and find where side b
meet the rulers. In this example, side b meets the ruler at 4-4 marks.
Therefore, area of p1: area p2 = 10:4 =2.5:1
p2p1
abA
1 2 3 4 5 6 7 8 9 10
10987654321
Compare the areas between the flags
Flags0
10'
9'
8'7'
6'
5'
4'
3'
2'
1'
10987
6
5
4
3
2
1
2
10=
1
5
Building a Sector Compass
Buy a pair of hinged rulers
Cut a wooden or cardboard model Hinge the pieces together Mark the rules with the correct marks
Overview
Description of device
Different scales
Lines of lines
Lines of superficies
Application
Building your own sector compass
Wood, F. (1954) Tangible arithmetic II: the sector compasses. The Mathematics Teacher. 12. p.535-541
Reference
Any Questions?
Coming Soon: www.math.utep.edu/alfredo/sector