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