You will need a sheet of Polar Coordinate paper; for practice, you can use the Low Resolution Polar Coordinate Graph; I will provide a high-quality graph, 12 inches square, for the actual project. This has concentric circles drawn at tenth-inch intervals from the center (the sheet I will provide is blown up a little, so the distance is a little bigger, but you will the distance from one circle to the next as being one tenth of an inch), and radial lines extending outwards at intervals as fine as 1 degree (near the outside of the paper), and as coarse as 10 degrees (near the center of the paper). Along the outside of the paper, pick one of the accented radial lines (either at the top or bottom) to represent 0 degrees celestial longitude (the direction of the Vernal Equinox), and mark it with a zero, then find the accented radial line 10 degrees counterclockwise from that (which should be the second accented line), and mark it with a ten, then continue, all the way around the graph, until you reach zero degrees again (this should give you numbers up to 350 degrees). Having done this, the center of the innermost circle (indicated by the intersection of two crossed lines) can be marked as being the position of the Sun. The various radial lines now represent different directions from the Sun, and the concentric circles would represent circular orbits at various distances from the Sun. (Note: This way of doing things is a scale model of the Solar System only if you were looking at the Solar System from the North side of our orbit. If you were to on the South side, the angles would have to go around clockwise, instead of counterclockwise).
      Note that you are supposed to be using pencil for all parts of this project. As you will see, you have to do a lot of drawing, erasing, and then redrawing in the last part of this project. If you use ink or make a mistake, you will have to get another sheet of graph paper, which I am willing to provide you, but which might not be available until the next time I see you, and will cost you 25 cents per extra copy.
      Your first step is to draw in the orbit of the Earth. On the scale that you will be using, the Earth's orbit is essentially circular, but you cannot use one of the printed circles, because they are all centered on the Sun, and the orbit of the Earth has its center offset from the Sun by a fraction of the orbital size called the eccentricity. For our orbit, this offset is about 1.5 million miles, and ignoring it would produce a similar error in parts of Mercury's computed orbit. Since Mercury's orbit is smaller than ours, these errors could be about 5% of the size of its orbit, which would be unacceptable.
      To find out how to draw in our orbit, look up the eccentricity and perihelion date for the Earth's orbit in a textbook, or encyclopedia, or calculate them directly from your data using the Extra Credit project, M3x. You will find that the eccentricity is about 1.7%, so that the offset is 1.7% of the radius of whatever 'circle' that you use to represent the Earth's orbit, and that the perihelion date is near the beginning of January. To find the position of the Earth, we use the fact that the direction we see the Sun is opposite (180 degrees different from) the direction the Sun sees us; so if you look up the Sun's longitude for each date (the same number you put next to each dot for the Sun), and subtract 180 (if possible; if the Sun's longitude is less than 180, add 180 to it, instead of subtracting), you'll get the longitude that the Sun sees the Earth at. For example, when we see the Sun at 280.3 degrees, it sees us at 100.3 degrees; and when we see the Sun at 9.0 degrees (88 days later), it sees us at 189.0 degrees.
      To draw the Earth's orbit, start with the largest complete circle printed on the graph paper (this would be 3 inches radius, if the graph were printed at its original scale, but will be around 6 inches radius, since I considerably magnified the printout). Draw in a new circle, the same size as the printed one, but moved in on the perihelion side of the Sun, and out on the aphelion side. This can be done by eye or with a compass, as discussed in class. The distance that the new circle has to be moved in is the eccentricity of our orbit (1.7%) multiplied by the radius of the circle, which is a little more than half-way from the pre-printed circle, to the next pre-printed circle on the perihelion side, and moved out the same amount on the aphelion side, but will lie exactly on the pre-printed circle at right angles to the perihelion and aphelion positions. (Note: The new circle will be shifted only halfway to the next NON-accented cirlce, or about a tenth of an inch, on the 12-inch graph paper. Do NOT move it in halfway to the next ACCENTED circle, which would be more than an inch, or you will have a terrible representation of the Earth's orbit.) Before going any further, show your new circle to the instructor, to make sure that you are doing this correctly.
      The position of the Earth on any given date can now be represented by finding the longitude of the Sun on that date, adding or subtracting 180 degrees to get the opposite direction, and then plotting a dot on the circle which represents the Earth's orbit, using the radial lines to find the angle which represents the appropriate position on that orbit. Plot dots in this way to represent the position of the Earth on the first 66 dates in the data table, and label every fifth dot (every 20 days) along the outside of the circle, with its Julian Date, so that you can easily find the position of the Earth on any given date. Again, if there is time to do so, show your work to the instructor before continuing.

      Look up the elongation which you measured for Mercury on the first date. Center a protractor (a 6 inch diameter protractor is the best size for this, ruled to the half-degree, so that you can estimate angles as fine as a 1/4 of a degree) on the dot which represents the position of the Earth on that date, and align (turn) it so that the 0 degree line on the protractor points exactly at the Sun (that is, the center of the graph). Use the radial lines on the graph to line up the 0 degree mark (its angular position, as measured by those lines, should be the same as the angular position of the dot for the Earth that it is centered on). Do NOT move the protractor so that the 0 degree line touches the Sun, as that moves the center of the protractor away from the dot which represents the Earth's position, and the protractor is only correctly positioned if the 0 degree line points at the Sun, AND its center is on top of the dot for the Earth. Make sure, before continuing, that the protractor is set up that way, so that both its center and its zero-degree line are properly positioned. Having done this makes the angles on the protractor equivalent to elongations. An angle 10 degrees counterclockwise from the zero degree line would represent 10 degrees East elongation, and one 10 degrees clockwise would represent 10 degrees West elongation. If you have a circular protractor, tape small pieces of paper on either side of the zero degree line, so that you can remember which side is East, and which is West. If you have a semi-circular protractor, there will be two zero degree lines. The one which has the angles running to its left (counterclockwise) should be marked with East, and the other one marked with West.
      Find the angle on the protractor which corresponds to the elongation of Mercry on the first date and, as accurately as possible, mark a small dot on the paper at the edge of the protractor. Use a straightedge to draw a line exactly through the Earth's dot, and the dot you just marked (note that you may have to hold the straightedge just to the side of the dots, so that the width of the pencil lead doesn't shove it past the dots), and extend it past the Sun, almost all the way to the other side of the Earth's orbit. If you have done everything correctly, Mercury must lie somewhere on this line. Otherwise, it would not be in that direction on that date.
      To find exactly where Mercury was on that line, you have to find the next time when Mercury, having completed exactly one orbital period, has moved all the way around its orbit, and is back in exactly the same place. Since the orbital period is 88 days, and your dates are separated by 4 days, this will be on the 23rd date. Although Mercury will now be in the same place that it was on the first date, the Earth is not, as we take a full year to orbit the Sun. Find the dot which represents the position of the Earth on the 23rd date, and align the protractor as before, so that the angles on the protractor represent elongations, then look up the elongation of Merucy which you measured for that date, mark the edge of the protractor with another small dot, and draw another straight line, through the new positions of the Earth and the marker. This will provide a completely different line, which will intersect the first line at a single point. If both lines were correctly drawn, Mercury must have been on each of them on the appropriate dates, but that is only possible if it was at the intersection of the lines, so that intersection must represent Mercury's position on both dates.
      Of course, there are many, many ways in which you might have made an error during or prior to this part of the project, so the intersection might not actually be the correct position. To see whether it is, find the position of the Earth, and the elongation of Mercury, on the NEXT date separated from the first two by a multiple of 88 days (this would be, in this case, the 45th date), and draw yet another line. Now, you should have three lines running across the graph, and if they are all correct, they will all meet at exactly the same point. If they do, mark that intersection with a small dot, and with some indicator of the first date that it was there, then erase all three lines except VERY close to the dot. You have to erase the lines, because they are only the first of 66 lines which you will draw, and if you leave them on the paper, it will soon be so filled with a maze of lines that you will not be able to tell what you are doing.
      If all three lines do NOT intersect at exactly the same point, their intersection will form a small triangle (if it forms a big triangle, you are doing something very, very wrong). This indicates that one or more of the lines is incorrectly drawn. Reposition the protractor for each of the three dates, and check to make sure that all three lines are drawn according to the Elongations you measured. If this does not reveal the problem, remeasure the Elongation of Mercury for all three dates, and try again. Usually, you will find that one of the three Elongation values was a little off, but sometimes, two or even all three of them need a slight adjustment.
      Once you have obtained a good result for the first trio of dates, go back to the second date, and 88 days after that, and 88 days after that, and draw another trio of lines, to find the position which Mercury had 4 days after it was at its first position. Once you have obtained a good result for this date, mark the dot, erase the new lines, and repeat the process, every 4 days, until you reach the 23rd date. Since you already used the data for that date in making the first trio of lines, if you were to do this again for that date, you would obtain the same dot as you did for the first dot, so there is no point in continuing, and you are almost done.
      The 22 dots which you have just created represent the position of Mercury every fourth day, throughout its orbit, and should form a perfectly smooth path. If they do, draw a smooth curve exactly through the center of every dot, to represent the orbit of Mercury, and show the result to the instructor. If, however, the path formed by the dots has any bumps or dips, or the spacing of the dots is uneven (they will gradually get further and further apart on the side of the curve which is closer to the Sun, and closer together on the other side of the orbit, but they should NOT be sometimes closer, then further, then closer again), it means that somehow, one or more of the dots, although apparently a good result, actually involved two or more lines being wrong, and just happening to intersect, anyway. In that case, go back to your original graph, remeasure the Elongations of Mercury for the dates involved, and redraw the trio of lines, to find the correct position. Once the graph looks nice and smooth, and the dots look nicely spaced, you can show it to the instructor for grading.
      There is one more part to this project, involving measuring this graph to see how well, or badly, it represents the orbit of Mercury, but at the moment, there are no copies available of the handout for that part, so I will have to discuss it in class, or individually, with those students who finish part 3b early enough to get to the last part.

Rough example of how to find the position of Mercury on January 1, 1982. On that date, the Earth is at 100.3 degrees longitude. The green and blue lines extending from that position to the Sun and to the left (counter-clockwise) of the Sun represent the Eastern elongation of Mercury on that date. Eighty-eight days later, the Earth is at 189.0 degrees longitude. The green and blue lines extending from that position to the Sun and to the right (clockwise) of the Sun represent the Western elongation of Mercury on that date. Eighty-eight days later, the Earth is at 274.0 degrees longitude. The green (not shown) and blue lines extending from that position to the Sun and to the right (clockwise) of the Sun represent the Western elongation on that date. If all three lines meet at a single point (as shown), that would be the position that Mercury had on all three dates, since they are separated by an orbital period. (This illustration is only an example of what things look like, in general. The lines are NOT drawn at the correct angles, which would be the angles you measured/calculated for Mercury, in part 2a; nor is their intersection the actual position of Mercury on that date.)