Lat. Am. J. Phys. Educ. Vol. #, No. #, Month, Year 1 http://www.lapen.org.mx

Eratosthenes’ measurement of the Earth’s radius in 1 a middle lab session 2 3 4

A. R. Mota1, J. M. B. Lopes dos Santos1 5 1 CFP e Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do 6 Porto, Porto P-4169-007, Portugal 7 8 E-mail: anaritalopesmota@gmail.com 9 10 (Received ####; accepted ####) 11 12 13

Abstract 14 In this article we describe a middle school lab session designed to explore and understand the Eratosthenes' method for 15 determining the Earth radius. The lab session is divided into six lab stations completely independent, though related, 16 each one with different apparatuses / materials. 17 The stations are diversified and range from simple tasks, like simulations or simple measurements to tasks that involve a 18 higher cognitive ability. The lab session includes several physical contents, such as the Earth movements (rotation, 19 translation and precession) and its consequences: variation of temperature, shadows length during the day, Sun rays 20 inclination, time zones and seasons. 21 22 23 Keywords: Earth movements (rotation, translation and precession), shadows length, Sun rays inclination, time zones, 24 seasons, Eratosthenes, Earth radius. 25 26 27

Resumen [Times New Roman 10] 28 Carefully review the wording of the abstract in Spanish. [Times New Roman 9] 29

30 Palabras clave: Up to three keywords should be supplied. Example: Conceptual errors on force, Physics Education 31 Include PACS numbers, Classical Mechanics teaching. [Times New Roman 9] 32 33 PACS: Insert at least 3 PACS classification numbers that are best suited. Example: 01.30.Os, 01.40.–d, 45.20.D– 34 Review: http://www.aip.org/pacs/pacs2010/individuals/pacs2010_regular_edition/reg00.htm#01. [Times New Roman 35 9] ISSN 1870-9095. 36

37 38

I. INTRODUCTION 39 40

The Eratosthenes’ measurement of Earth’s radius is one 41 of the most popular and fantastic experiences in the history 42 of Science. Over the decades, it has always been admired 43 not only because of its apparent simplicity but also due to 44 the different domains of knowledge that it involves: 45 Geography, Mathematics and Physics. 46

In spite of some of the measurements involved in this 47 experience being subject to several inaccuracies, is 48 remarkable the procedure and the final result, closer to the 49 correct one. 50

Unfortunately, students are not always aware of this 51 historical treasure; probably perhaps teachers have 52 difficulties in its adaptation to the level they teach. 53

Let’s remember the story: Eratosthenes had heard that in 54 the summer solstice (the longest day of the year), at noon, 55 the Sun was straight above the town called Syene (modern 56 Aswan). The travelers reported that a vertical pole cast no 57 shadow at that time in that town. 58

In Alexandria, where Eratosthenes lived, he observed 59 that in the same day, at the same time, the Sun cast a 60

shadow. He measued the angle of the Sun rays in that 61 conditions, and according to Kleomedes, Eratosthenes 62 obtained 7º 12' (1/50th of a circle, e. g. 7.2º). 63

Observing a map of ancient Egypt is relatively easy to 64 see that Syene is closer to the Tropic of Cancer and 65 Alexandria and Syene lay apparently on a direct north-66 south line. 67

At that time, the distance from Alexandria to Syene was 68 known to be about 5000 stadia. Eratosthenes' original 69 writings on the measurement of the Earth were lost, 70 although legend has it that he had someone walk from the 71 two towns. It is hard to know for sure what corresponds 72 exactly the unit stadia. Despite the controversy discussion 73 about the ancient units1, it is more or less consensual that 74 one stadia would correspond to one-tenth statute mile1. In 75 this case, 5000 stades would correspond at 500 miles (about 76 804.6 km). 77

The Fig. 1 and the geometric principle that opposite 78 interior angles of two parallel lines are the same help us to 79 understand how Eratosthenes has determined the Earth 80 radius. Thus, the angle subtended by the arc between Syene 81 and Alexandria at the earth’s center is the same as the angle 82

A. R. Mota, J. M. B. Lopes dos Santos

Lat. Am. J. Phys. Educ. Vol. #, No. #, Month ### 2 http://www.lajpe.org

of the Sun rays inclination at Alexandria (angular deviation 1 of the sun from the vertical direction in the Alexandria 2 surface). Moreover, once the distance between Alexandria 3 and Syene is 1/50 of the circumference of the Earth, we can 4 readily deduce: 5 6

r ! 804600 "502!

(1) 7

8

9 FIGURE 1. The calculation of the earth radius by Eratosthenes 10 11

Clearly, this method was based on a proportion relating 12 the difference between the latitudes of two locations and 13 their terrestrial distance apart2. Moreover, it was developed 14 on the assumption that the Earth is spherical and that the 15 Sun is so far away that its rays can be taken as parallel. 16

Nowadays we know that this straightforward 17 methodology contais some false assumptions and flaws. 18 However, it is important to stress that Eratosthenes did not 19 have an accurate knowledge of geographical positions, so 20 any measurement given by him is conditioned even before 21 any type of result. 22

Eratosthenes assumed erroneously that Syene was 23 exactly on the Tropic of Cancer and indeed it is 37 miles to 24 the north of the Tropic. He believed also that Syene and 25 Alexandria were on the same meridian. In fact, Syene is 26 about 3º east of Alexandria. 27

Moreover, the true distance between Alexandria and 28 Syene is 453 miles and not 500 previously mentioned by 29 Eratosthenes and the difference of latitude between them is 30 7º7’, rather than 7º12’ as he had concluded. 31 32 II. Procedure 33 34

It was designed a middle school lab session, specifically 35 for the 7th grade, to explore and understand this historical 36 experience. The lab session was held along two lessons, 37 each one lasting 90 min. 38

Each lesson was divided in a lecture session and a lab 39 session. The contents for each lesson are in Table I and II. 40

The lab session was designed around an interactive 41 model of laboratory classes that combine hands-on- 42 activities in an active learning environment: the lab stations. 43

44 45

TABLE I. Contents and lab stations for the first lesson. 46 47

Lesson Contents Lab stations

1

Earth’s movements;

Variation of temperature;

Time zones

1

2

48 TABLE II. Contents and lab stations for the second lesson. 49

50 Lesson Contents Lab stations

1

Sun’s rays’ inclination

Shadow’s length during the day;

Seasons (introduction)

The radius of the Earth by Eratosthenes

1

2

3

4

51 Altogether the session had six lab stations that we will 52

describe, after explaining the concept of this model. 53 54 III. LAB STATIONS 55 56

Laboratory activities play a central role in the learning 57 of physics regardless of the educational level3. However, 58 despite their proven advantages, teachers avoid them 59 (preferring demonstrations) because they require a 60 significant amount of material, take a long time to prepare 61 and to implement. 62

In this model, the classroom / laboratory is divided, 63 usually, into few experimental stations, each one with 64 different apparatuses / materials and students, divided in 65 groups of three or four, travel from station to station with a 66 pre-defined time frame. 67

The lab stations must be independent because each 68 group has its own sequence. 69

The stations may be diversified and can range from 70 simple tasks (such as simple measurements) to tasks that 71 involve a higher cognitive ability, like testing hypotheses, 72 constructing explanatory models from observations or 73 solving open-ended problems. 74

At every station, students also have to answer some 75 theoretical questions from a worksheet: the lab worksheet. 76 The teacher serves as a supervisor, being aware of the 77 students’ difficulties, monitoring the discussions within 78 groups and thus providing a personalized learning. 79

The lab stations have many advantages when compared 80 with typical lab sessions; a reduction of the material 81 requirements, a formative assessment, a better work flow 82 from students and a higher motivation. 83 The higher variety of experiments and the possibility to 84

develop different types of abilities, like conceptual and 85 quantitative understandings of physics principles and 86 science process abilities also contribute for the advantages 87

Eratosthenes’ measurement of the Earth’s radius in a middle lab session

Lat. Am. J. Phys. Educ. Vol. #, No. #, Month ### 3 http://www.lajpe.org

of this type of sessions. Note that this type of sessions is 1 adaptable to any syllabus and can provide a personalized 2 learning. 3

All the material is stored in kits properly identified to 4 facilitate the subsequent use, so teachers spend less time in 5 preparing and storing the material. In each kit, the material 6 is organized by lab station (Fig. 2). 7

8 9

FIGURE 2. Kit with the material required, organized by lab 10 station 11

12 The lab session presented can be easily constructed by 13

teachers with few and low-cost materials. All of the lab 14 stations lasting 10 min. 15 16 A. LAB STATION 1 (1st LESSON) 17 18 The first two lab stations were implemented in the first 19 lesson, after the lecture session. An overhead projector was 20 available; in order to represent the Sun. There were also a 21 thermometer, a globe and some flags indicating several 22 countries (Fig. 3). 23 24

25 26

FIGURE 3. Picture of the Lab station 1 (1st lesson). 27 28 The questions and instructions related to this station are 29 described as following: 30 31 1. Look at the flags in the earth globe. In which countries 32

is day time? 33 2. In which country did the sunset occurred first? 34 3. In which country did the sunrise occurred first? 35 4. Indicate the cardinal points (north, south, east, west) of 36

each country represented, in relation to Portugal. 37 5. Measure the temperature of a country where it is still 38

day time and of another country where it is night. What 39 conclusions can you draw? 40

41 B. LAB STATION 2 (1st LESSON) 42 43

This station is related with the Earth movements and 44 time zones. The students had at their disposal many small 45

size globes (Fig. 4) to simulate the three movements 46 learned (rotation, translation and precession) with their 47 peers. Then, they simulated the three movements in the 48 teacher’s presence, for assessment. In the second part of the 49 lab, the students made a prediction time in some 50 international locations (taking Lisbon time as reference) 51 with the help of a normal size globe and a time zones map. 52 53

54 55

FIGURE 4. Picture of the Lab station 2 (1st lesson). 56 57 Here are the questions and instructions: 58 59 1. Indicate the Earth movements and simulate them for 60

your teacher. 61 2. If it is the teatime in London, the famous Big Ben strikes 62

5 p.m.. What time is it in Paris? 63 3. If it is 10 a.m. in Lisbon, what time is it in... 64 a) ... London? 65 b) … Azores? 66 c) ... Brussels? 67 4. In 2008, the Olympic Games were held in Beijing. The 68 Portuguese triathlon champion and five-times champion of 69 Europe, Vanessa Fernandes, held its race at 10 a.m. (local 70 time). What time was in Lisbon when Vanessa started the 71 race? 72 73 C. LAB STATION 1 (2nd LESSON) 74

This lab station was designed to study in a real situation 75 context the shadows length during the day. The overhead 76 projector represented the sun and a small stick (made by 77 rubber) was placed on the globe, in Portugal (Fig. 5). 78

More specifically, the questions and instructions given 79 were: 80 81 82

83 84

FIGURE 5. Picture of the Lab station 1 (2nd lesson). 85 86

1. Put the small stick (made by rubber) in Portugal. Turn 87 on the overhead projector to simulate the sunrise in 88 Portugal. 89

A. R. Mota, J. M. B. Lopes dos Santos

Lat. Am. J. Phys. Educ. Vol. #, No. #, Month ### 4 http://www.lajpe.org

2. How does shadow length changes during the day? 1 (Simulate with the globe) 2

3. If you are facing your shadow when its length is 3 minimal, which is the cardinal point that you see ... 4

a) in front of you? 5 b) behind you? 6 c) on your right? 7 d) on your left? 8

9 10

D. LAB STATION 2 (2nd LESSON) 11 12

The concept of shadow and its main features is studied 13 in this station. Here, the students explored the size of a 14 shadow by moving an object closer or farther to the light 15 bulb and, as the source has an appreciable size, they could 16 see the two distinct regions of a shadow; one of full-17 shadow, called the umbra, the other of half-shadow, called 18 the penumbra (the penumbra doesn't have a uniform 19 shadow density, but rather a gradient). 20

At the end, the students must draw a geometrical sketch 21 where they have to identify the two regions. This station 22 was designed thinking in the next lessons where the 23 students would learn the eclipses. Here are the questions 24 and instructions in the lab worksheet: 25 26 1. Why do shadows appear? 27 2. Explore the size of a shadow by moving an object closer or 28

farther to the light bulb. What conclusions can you draw? 29 3. Make a geometrical sketch where you can identify the umbra 30

and the penumbra. 31 32

33 34

FIGURE 6. Picture of the Lab station 2 (2nd lesson). 35 36 37 E. LAB STATION 3 (2nd LESSON) 38 39

The second lesson here presented preceded the study of 40 the seasons. Thus, it was intended that students study the 41 Sun rays inclination before having any formal instruction 42 about this subject. To explore this idea, a flashlight and a 43 paper sheet with a square drawn on it were available. The 44 main goal was that students recognized by their own 45 experience that since they raise the inclination, the same 46 energy is distributed over a larger area. Thus, the greater 47 the slope, the lower the energy per unit area focused. 48

49

50 51 FIGURE 7. Picture of the Lab station 3 (2nd lesson). 52 53 More specifically the questions and instructions were: 54 55 1. In which situation the square takes more energy? 56 2. In which situation the sheet takes more energy? 57 3. In which situation there is a higher inclination of the 58

rays emitted by the flashlight? Justify. 59 4. Which situation can represent the summer? Justify. 60 5. You have two globes in two situations (A and B) and a 61

flashlight in a stand. Turn on the flashlight and see the 62 illuminated area in each case. Focus your attention in 63 the same region, for instance, Portugal. 64 a) In which case was less energy per area? 65 b) Which situation can represent the summer? 66

67

68 69

FIGURE 8. This picture shows the globe representing the 70 summer. 71

72 73 F. LAB STATION 4 (2nd LESSON) 74 75

The Eratosthenes' method for determining the Earth 76 radius is studied in this lab station. Is advised to talk about 77 this experience in the lecture session, preferably after the 78 concept of Sunrays inclination. The Fig. 1 must be 79 presented previously as well as the geometrical exploration. 80

In their workstation, students had a globe, an ancient 81 Egypt map and a protractor. 82

In the beginning of this station, the students read a text 83 with the experience described. Then, they had to answer the 84 following questions and carry out some instructions: 85 86

Eratosthenes’ measurement of the Earth’s radius in a middle lab session

Lat. Am. J. Phys. Educ. Vol. #, No. #, Month ### 5 http://www.lajpe.org

1 2

FIGURE 9. Picture of the Lab station 4 (2nd lesson). 3 4 1. Explain the absence of shadow in Syene in that specific 5

day. 6 2. With a flashlight illuminate the city of Syene in order to 7

demonstrate the summer solstice. 8 9 10

11 12 FIGURE 10. Picture of a student answering the question 2. 13 14 3. In a regular day (not in summer solstice), Syene and 15

Alexandria have shadows. Will they be of the same size? 16 4. If the Earth were flat, the shadows length in Syene and 17

Alexandria, at the same time, be the same? 18 5. In the following diagram there is a representation of the 19

experience of Eratosthenes. In this diagram... 20 21 22

23 24

FIGURE 11. Diagram from the question 5 25 26

a) … draw the sun ray. 27 b) … find the angle between the Sun and the vertical 28

direction. 29 6. Calculate the Earth radius through this experience. 30 31 VI. CONCLUSIONS 32 33

The methodology implemented in this session allowed 34 students an active learning and a reasoning understanding 35 of some important physical concepts. 36

The lab stations were crucial to make students 37 understand not only the Eratosthenes' method for 38 determining the Earth radius, but mostly to deal with daily 39 problems by observing and touching, being aware of some 40 particular physical details. 41

Undoubtedly, there is a gap between seeing and 42 understanding, although, it will depend on how laboratory 43 lesson is staged and intellectual activity is organized around 44 the experiment4. At this rate, the lab stations were not 45 independent hands-on activities; all of the steps were 46 important ingredients designed in a reasoning line of 47 questioning. 48

The presented pedagogical approach to the 49 Eratosthenes’ measurement of the Earth’s radius help 50 students to construct gradually their knowledge by 51 themselves, but, it is necessary that this type of teaching 52 strategy continue with other subjects during the all year. 53

The assessment was an important tool of this lab 54 session. The teacher could see in the lab all the discussions 55 within groups and intervene with more questions, in order 56 to promote conceptual change and a reasoning learning. 57 The lab worksheet was also evaluated, so the teacher could 58 make aware of the main students’ difficulties. 59

The students enjoyed this session and felt more 60 motivated. However, they argued a short time for doing 61 each one of the labs. In some cases, they asked more time 62 in order to make some demonstrations slower and to have 63 more discussion within group. The answers, according to 64 the students, would be more complete and consolidated. 65 66 67 ACKNOWLEDGEMENTS 68 69

This work was supported by the European Union, 70 program POCI 2010 through project Ciência Viva PVI/252. 71 72 73 REFERENCES 74 75 [1] Engels, D., The Length of Eratosthenes’ Stade, The 76 American Journal of Philology 106, 298-311 (1985). 77 [2] Dutka, J., Eratosthenes' measurement of the earth 78 reconsidered, Archive for History of Exact Sciences 46, 55-79 66 (1993). 80 [3] E. Etkina, A. Karelina, M. Ruibal-Villasenor, How long 81 does it take? A study of student acquisition of scientific 82 abilities, Phys. Rev. ST Phys. Educ. Res., 4(2), 0201108 83 (2008). 84 [4] Viennot, L., Teaching Physics (Kluwer Academic 85 Publishers, Dordrecht, 2003). 86 87 88