# 1 Mile = ???

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1 mile = how many minutes??

I know this is probably a very elementary question but this feeble mind would like to know.

Similar questions have been asked, but usually the question is how many feet is in a degree or minute.

Since longitude lines are parallel, there will be a consistent Distance-Degree ratio, but with latitude the distance varies depending on how far North or South you are.

With that in mind:

1 mile longitde = .01446 degrees or .867 minutes.

1 mile latitude (at ~0º longitude) = .01446 degrees or .867 minutes.

1 mile latitude (at ~45º longitude) = .02039 degrees or 1.224 minutes.

These calculations can be easily done with a GPSr by making waypoints with the same latitude or longitude and calculating the distance or degrees difference.

I have gone to look for myself. If I should return before I come back, tell me to wait here until I return.

[This message was edited by brdad on August 08, 2002 at 06:39 AM.]

quote:

These calculations can be easily done with a GPSr by making waypoints with the same latitude or longitude and calculating the distance or degrees difference.

Careful with that; it might not quite work like you expect. If your GPSr computes great circle distance - and it probably does - you'll get a slightly shorter distance between two points at the same latitude than the distance you'd walk if you followed the latitude (rhumb) line. For two coordinates a minute apart, the difference is probably negligible, but it's there.

Warm Fuzzies, you're right. I knew of the difference, but didn't mention it because I figured GPSr would be close enough and I'll admit I wasn't sure just how much of a difference it would make.

I just did a basic calculation and came up with about a 27 mile difference between great circle and rhumb line at the equator halfway around the earth - not nearly as big a difference as I thought. Not sure if you've ever looked into it or not. I'll go and verify my calcultions and post if I made a mistake.

I would assume a simple GPSr uses the great circle distance too, be intersting to know that for sure too...

Note: I was able to come up with almost 500 mile difference on very long distances, much bigger difference than 27 miles and closer to what I was thinking the difference was.

I have gone to look for myself. If I should return before I come back, tell me to wait here until I return.

[This message was edited by brdad on August 08, 2002 at 07:44 AM.]

[This message was edited by brdad on August 08, 2002 at 07:58 AM.]

quote:

I just did a basic calculation and came up with about a 27 mile difference between great circle and rhumb line at the equator halfway around the earth - not nearly as big a difference as I thought.

The equator is both a rhumb line and a great circle, though, so your difference might have more to do with eccentricity. Try it at a latitude other than 0 and the difference might be more noticeable.

KS

Your question is not as elementary as you may think. Strictly speaking, there is no exact answer because the earth is dynamic, always changing in size and shape. Furthermore, the earth is not perfectly spherical, so any answer would be only an approximation. Science has developed many mathematical models over the centuries, approaching the true magnitude of the earth ever closer as measurement technology improves. This is what accounts for the changes in the coordinates and elevations found on the data sheets over the years. Also, it should be understood that miles are units of distance and minutes are units of angle, so there is no absolute equivalency between the two, as there is with feet-to-miles or minutes-to-degrees for example.

Treating the earth as a sphere for simplification purposes, one must determine the radius of the earth. My high school textbook said it was considered to be about 20,900,000 feet at that time, so I have used that number.

If you wish to compare miles along the surface of the earth to minutes of latitude, which are measured north and south starting from the equator, imagine a circle passing through the poles. The circumference of that circle is about 131,318,500 feet or 24,870 miles. There are 360 degrees in a circle so there are about 69 miles per degree. There are 60 minutes per degree so 1.15 miles equals one minute of latitude. This is true no matter where you are on the earth, either near the equator or near the poles, because lines of latitude are parallel.

If you wish to compare miles to minutes of longitude, which are measured east and west from the Greenwich Observatory, it gets a bit more complicated, because lines of longitude are not parallel. Along the equator, the numbers are identical to those given above, so the result is the same 1.15 miles per minute. The radius of each circle of latitude diminishes as you move away from the equator, so the length of a minute of longitude also diminishes, to zero at the poles.

At 45 degrees latitude, which is often used as the mean for North America, The radius is about 14,778,500 feet. The circumference of that circle is about 92,856,000 feet or 17,586 miles. Divide by 360 degrees and you see that there are only about 49 miles per degree of longitude at this point on the earth's surface. Divide by 60 and you get 0.81 miles per minute of longitude at 45 degrees of latitude, either north or south of the equator.

The subject of measurements on the surface of the earth is known as Geodesy. A search for this subject will reveal a virtually infinite amount of further info, if you wish to pursue it.

Ah, life would be so much simpler if Columbus had been wrong and the earth was flat...

I'm still curious about how the GPSr navigates - When in navigation mode does the pointer point to the waypoint and calculate distance to goal in a rhumb line or great circle? Probably the better ones can do both, but I'm curious about basic.

Warning: Objects in GPS may be closer than they appear!

[This message was edited by brdad on August 08, 2002 at 07:29 PM.]

I'm not sure I understand the way you guys are using the word rhumb. To my understanding, every parallel of latitude is rhumb with respect to every meridian of longitude it crosses, and vice versa, so I don't know what other line or difference you are talking about. Perhaps you are thinking of the difference between the arc distance along the surface and a chord distance, which would actually be running underground directly from point to point.

I assume the Great Circle Calculation is used (which is what we use on the Geocaching.com site). Just on the basis of simple=better for GPS units, and accuracy on some level can be forgiven, given the discrepancy in 6 meter accuracy.

FYI, Conversions from WGS84 to NAD27 on at least the Garmin units are done with the Molodensky transform method. I'm trying to sort this one out now so I can provide accurate links to Topozone maps. I just figured out WGS84 to UTM which is a huge triumph. I love this Dot Net stuff.

I also never expected to be a GIS guy. Crazy world we live in.

Jeremy Irish

Groundspeak - The Language of Location

survey tech - this is my very basic knowledge of great circle versus rhumb line. I learned more yesterday than I ever knew, but this basic explaination that I leaned a few years back still holds true.

Imagine travelling from the West to the East coast , say 45ºN 123ºW to 45ºN 68ºW Assuming flat terrain and no obstacles, and travelling true East (therefore travelling along the 45ºN line all the time), that would be travelling the rhumb line for 2694 miles.

Now, the way I was shown the great circle line is if you took those same coordinates on a globe, and stretched a string tight beween them, the shortest course would not be the rhumb line, but where the string was, the great circle line, which I calculate to be about 2640 miles, for a difference of 54 miles.

One can see quickly that if you were near one of the poles and wanted to go 180º longitude from where you were, that the shortest course would be right over the pole, not due East or West.

Because the longitude lines converge at the poles if you are travelling true North or South there is no difference. Travelling the equator there will be no difference either as I was reminded of yesterday. Any other direction will yeild some discrepency.

Now maybe someone can tell me if I passed that test?

Warning: Objects in GPS may be closer than they appear!

[This message was edited by brdad on August 09, 2002 at 03:45 AM.]

survey tech - this is my very basic knowledge of great circle versus rhumb line. I learned more yesterday than I ever knew, but this basic explaination that I leaned a few years back still holds true.

Imagine travelling from the West to the East coast , say 45ºN 123ºW to 45ºN 68ºW Assuming flat terrain and no obstacles, and travelling true East (therefore travelling along the 45ºN line all the time), that would be travelling the rhumb line for 2694 miles.

Now, the way I was shown the great circle line is if you took those same coordinates on a globe, and stretched a string tight beween them, the shortest course would not be the rhumb line, but where the string was, the great circle line, which I calculate to be about 2640 miles, for a difference of 54 miles.

One can see quickly that if you were near one of the poles and wanted to go 180º longitude from where you were, that the shortest course would be right over the pole, not due East or West.

Because the longitude lines converge at the poles if you are travelling true North or South there is no difference. Travelling the equator there will be no difference either as I was reminded of yesterday. Any other direction will yeild some discrepency.

Now maybe someone can tell me if I passed that test?

Warning: Objects in GPS may be closer than they appear!

[This message was edited by brdad on August 09, 2002 at 03:45 AM.]

Looks to me like you passed it. The shortest explanation of the difference I know is that a great circle is an arc of the largest circle you can draw through the two points while remaining on the surface of the globe, and it is always the shortest distance between the two points (in math geek terms, a great circle is a geodesic). A rhumb line is the curve you follow if you make sure your heading stays constant through the whole trip.

Rhumb lines are straight lines on a Mercator projection, so stretch a string between the two points on a wall map to see the rhumb line route. Great circles are "straight" (geodesic) lines on a globe, so stretch a string between the two points on a globe to see the great circle route (or look at the various shipping routes on your wall map; they tend to be great circles.)

All latitude lines are rhumb lines (constant heading of either 90 or 270) and all longitude lines are rhumb lines (constant heading of either 0 or 180.) Longitude lines are also great circles, as is the equator. Other latitude lines are not great circles.

I see, that difference would obviously be negligible over any distance short enough to be covered on foot.

when I set a waypoint on my latitude but at 0 degrees longitude some time back, my unit did not display 90 degrees to the waypoint, but some other heading (obviously using the Great Circle route).

Here is a table the is helpful for showing differences in meters in different degrees longitude as you go up the latitude scale:

http://home.online.no/~sigurdhu/Grid_1deg.htm

quote:
I also never expected to be a GIS guy. Crazy world we live in.

Who would have thought that finding two (!) lousy benchmarks with my best friend out along the railroad tracks (where we were not supposed to be) when we were 12 would lead to such an interest in tracking down so many others and learning how and why they are there. Crazy world indeed!

quote:
Originally posted by EraSeek:

when I set a waypoint on my latitude but at 0 degrees longitude some time back, my unit did not display 90 degrees to the waypoint, but some other heading (obviously using the Great Circle route).

Yep, you are right. I tried that myself, so that pretty much confirms a GPSr would use great circle calculations. But try changing the latitude to 0 instead and you'll get 180º (true, not magnetic), or 0º if you were below the equator.

Warning: Objects in GPS may be closer than they appear!

rotflmao! sorry, i went out to happy hour with co-workers after work today!

Actually at 37 degrees longitude one degree = 55.4 miles, one degree of lattitude =69 miles the average of (88 feet+-) = 1 second of time has worked real well for me. do not have the precise breakdown readily available but you can do the math how precise is precise. 1.1 nautical mile = 60 seconds I believe. thus the Rectangular Survey System of the Public Lands

quote:
Originally posted by Trailblazer # 1:

1 mile = 5280' divide by 60 seconds =( 88' +-)

Took me a minute to soak that one in! Looked like you were trying to state that 1 mile = 88 feet.

Anyway, average figures that work at one latitude may be a fair amount off at another latitude. At 80º latitude 1º longitude is only equal to .1748 nautical mile (1062 feet) Check out the chart EraSeek mentioned above.

At 80º latitude 1º longitude is only equal to 10.5 nautical miles and 1 minute is equal to .1748 nautical mile (1062 feet) Check out the chart EraSeek mentioned above.

It really does not matter for geocaching, or for that matter anything most of us would be doing with our GPSrs. Obviously when you get into professional surveying, or long distance trips on flat terrain (like sailing or flying) then it could make a big difference.

Warning: Objects in GPS may be closer than they appear!

[This message was edited by brdad on August 10, 2002 at 01:20 PM.]

T1

The Rectangular Survey System of the Public Lands uses parallels and meridians, but is not related to latitude and longitude in any way. The Townships, which, in theory, each consist of 36 sections, each of one square statute mile (5280 feet), not nautical mile (6080 feet), are laid out across each state or region from an initial point, which is random with respect to latitude and longitude. So the corners of the sections are all at random lat/long locations and there is no equation by which the two systems can be related.

Say, Paul Bunyon, I think you meant one minute, not one degree, of long. = 1062 feet at 80 degrees lat., right? The chart is fine, but it uses meters and nautical miles instead of feet and statute miles, which are normally used in this country, so it introduces the need for additional conversions.

Thanks survey tech, I edited it to reflect the right numbers the way I had intended.

Maybe I'll play with the idea of making a new chart using miles and an explaination of what we're talking about, seems someone is always asking.

Duh, maybe I won't make one. After looking at that page again, I noticed they had a link using imperial units: ericcloninger.com

Warning: Objects in GPS may be closer than they appear!

[This message was edited by brdad on August 10, 2002 at 02:05 PM.]

[This message was edited by brdad on August 10, 2002 at 02:06 PM.]

That being at 37 degrees longitude. The present system of Governmental Land Surveys adopted by Congress 7 May, 1785 "Rectangular System", that is, all its distances and bearings are measured from Two Lines which are at right angles to each other,viz:+. These Two lines,from which the measurements are made,are the Principal Meridians, which run North and South,and the Base Lines that run East and West. Is this not Longitude West from Greenwhich? you can as what another person suggested for the measurement put way points as true lines of Long. and Lat. ie. 4 points 93 45 00 36 00 00 , 93 46 00 37 00 00 ,93 45 00 37 00 00 , 93 46 00 36 00 00 i think i got that right i am sure if its not you'll catch it!

The initial point was made by astronomic observations on a true line of longitude or the local hour angle, prior to the adoption of the Greenwhich hour as the standard ,local surveyors made their observations on their local hour angle,and astonomic observations for lattitude angle much like the GPS automaticaly does for us, the older surveyors had to be very knowledgeable in the locating of lines. if you mathmaticaly start from a true meridian Township 1 Range 1 6 miles by 6 miles less the convergency of the meridians you will find interesting math coorealations . thes will also correspond to bench marks and triangulation stations. I am still working on a few bugs of my own i have had to study and learn alot of this on my own. And am still trying to learn more!

Yep, just when you think you know it all, you realize you know nothing!

As I said in my third previous post, life would be so much simpler if Columbus had been wrong and the earth was flat... Actually, I'd be happy if magnetic north would just move to the north pole!

Makes you wonder how they even got it anywhere near as accurate as they did years ago. I have a customer that's 95 or so, when he was in mid 20's he surveyed for 2 years up in Eastern Canada all four seasons - some good stories there!

Warning: Objects in GPS may be closer than they appear!

T1

The Initial Points that control the Rectangular System were set at arbitrary locations, chosen because they were considered convenient, with no regard for lat/long. For example, the Initial Point that controls your state of Missouri, and several others, is located at 38-38-45 / 91-03-07, which as you can see, is a random location with respect to lat/long. Once again, there is no connection between the Rectangular System, which governs property lines, and the NGS control network, which serves engineering and mapping purposes. In fact, they are not even in the same branch of government. The rectangular system is in the Department of The Interior and the NGS is in the Department of Commerce. Get a copy of "The Manual of Surveying Instructions for the Survey of the Public Lands of the United States", last revised in 1973, to learn more.

Not hard to guess when there is more than one system involved, and neither one is remotely related, the government must have designed it!

Warning: Objects in GPS may be closer than they appear!

Since the times of the earliest surveys, the townships and sections have been located with respect to Principal axes passing through an origin called an initial point;the North-South axis is a true meridian called the Principal Meridian, the 5th Principal for Missouri, and the East-West axis is a true parallel of lattitude called the base line. This is True lines 91 00 00 38 00 00, ect. is it not?, which are logitude and lattitude if you base these measurements from the Prime Meridian being Greenwhich 00 00 00 , according to the GLO or General Land office book 1906, and the analysis of the United States Surveys,same book and the Principal Meridians,Base Lines, and areas Governed there by,same book entered into the Library of Congress, which I have in my posession show these type measurments,I also have the 1973 BLM survey Manual as well as many other survey books. the key map from the old book is the best example of them all. It actually shows all lines in Missouri as well as the Principal Meridians and Base lines for the entire USA. thanks for keeping my mind working in this, there are still several unanswered questions

The reading 38 38 48 91 03 07 as the Initial point according to my calculations throw the Range Lines off by a couple of miles according to the usgs maps datum, maybe I am wrong!!! if you start Township 1 Range 1 from this reading. I have several of the Townships, and sections 22,23, Range 25-26, 26-27 27-28,corners recovered, these can be as fun as finding benchmarks as well, and all the witness corners ,trees, and the monuments associated with them

Yes, thats what I have been trying to tell you. Range lines will not match up with any meridians of longitude because they do not correspond to each other. The Rectangular System was intended to create square sections. This would not be possible using meridians of longitude because they are not parallel. The meridians of the Rectangular System have a jog every 24 miles as a way of correcting for convergence. Refer to page 62 of the 1973 manual. The two systems are similar but not the same.

From this initial point then, why are all the monumentations of the Townships,not in order?, That is 6 miles by 6 miles if you begin from this initial point then the Range Lines or Township lines should correspond, Range 1 should be 6 miles,Range 2, 12 miles ect. these are the Range lines.Maybe I was wrong in the way I presented the issue in previous discussions.

I posted this somewhere in July 2001. It still holds.

Figure the numbers of feet per 1/1000 minute of latitude and longitude as follows. The number is always 6.07 ft per 0.001 minute of latitude, everywhere. However the feet per 0.00l minute of longitude is always smaller than this by a factor of the cosine of your latitude. This is all explained so:

A good number to use for the mean radius of the Earth is 3955 miles. That is 3955 (mi)* 5280 (ft/mi) = 20,882,342 (ft) radius. The circumference is 2 pi times that, or 2 * 3.14159 * 3955 / 5280 = 131,207,627 (ft). There are 360 degrees around the circumference, and 60 minutes per degree. Therefore there are 131,207,627 (ft/circ) / 360 (deg/circ) / 60 (min/deg) = 6,074 (ft/min).

And so, there are 6,074 / 1000 = 6.07 ft per 1/1000 of a minute.But this is for a minute of angle around a great circle, like the equator, or a latitudinal meridian line. So, since all meridians are great circles encircling the Earth in N - S planes, when figuring latitudinal distances, a thousandth of a minute of latitude is everywhere the same; it is 6.07 (ft / 0.001 min lat).

When figuring longitudes, however, since the circles of longitude (which encircle the Earth in E - W planes parallel to the equator) are smaller and smaller as the latitude approaches the pole, the number of feet in the circumference of those smaller circles is always less than that in a great circle like the equator.

How much smaller a minute of longitude is than a minute of latitude is merely the cosine of the angle of latitude. You can get this cosine on any scientific pocket calculator or from a table of trigonometric functions.

Here in southern Michigan, where the latitude is about 42 degrees, since the cosine of 42 degrees is 0.7431, the number of feet in a thousandths of a minute of longitude here is calculated as 6.074 (ft/0.001 min at equator) * 0.7431 (cosine of 42 deg) = 4.51 (ft/0.001 min at 42 deg).

At some other latitude where you may be, the number of feet per thousandths of a minute longitude equals 6.07 times the cosine of your latitude.

Aside from the variance in the radius of the earth, your explanation is nearly identical to mine, although you stated it more eloquently and made reference to the use of the cosine function, which I neglected to mention. Just to clarify for those taking notes, in the fourth paragraph "latitudinal meridian line" should read "meridian of longitude" and in the fith paragraph "circles of longitude" should read "parallels of latitude". Well stated, that should clear this up once and for all for everybody.

A minute is a unit used to measure both time and angle. World Book Encyclopedia (Minute) A circle is divided into 360 degrees, so one minute is 1/21,600 of a complete circle.Each minute of an angle is divided into 60 seconds. The minute in time is an exact measurement, which means exactly so much time.The minute of an angle is an exact portion of a circle, and is independent of the size of the circle. But if the angle is denoted by a linear measurement along the circumference of the circle,the distance of a minute depends on the diameter of the circle. On the Earths surface it is 1 (one) nautical mile, about 6,076 feet.See Mile World Book Encyclopedia Degree, Because Longitude and Lattitude lines are circles, they are also measured in degrees.so if you use the figure 1 Minute = (60 seconds) = 1 mile @ 6,076 feet divided by 60 = 101.266'per second of time. 1 minute = (60 seconds)Land Measure Mile 5280' divided by 60 = 88' per second of time. These are just my observations , using the 1906 (GLO) Plats, Survey of all Missouri, Geodetic Scale 37-38 Degrees 1:31680 Scale GPS and Monumentations (Bench Marks from 1909-2002)

If you program a true line of Lattitude from say 93 00 00 , 37 00 00 to 00 00 00, 37 00 00 you come up with the angle of approximately 57 degrees +-which has to do with radian measurement which has no arbitrary units, if you watch the space shuttle lift off angle it is 57 degrees? I am still learning about this type of system so no quotes please maybe one of the smarter members might have something to say on this one

quote:
Originally posted by survey tech:

T1

The Rectangular Survey System of the Public Lands uses parallels and meridians, but is not related to latitude and longitude in any way. The Townships, which, in theory, each consist of 36 sections, each of one square statute mile (5280 feet), not nautical mile (6080 feet), are laid out across each state or region from an initial point, which is random with respect to latitude and longitude. So the corners of the sections are all at random lat/long locations and there is no equation by which the two systems can be related.

Say, Paul Bunyon, I think you meant one minute, not one degree, of long. = 1062 feet at 80 degrees lat., right? The chart is fine, but it uses meters and nautical miles instead of feet and statute miles, which are normally used in this country, so it introduces the need for additional conversions.

When the township system was surveyed for the state of Montana, it worked out, roughtly, so that the western, rugged mountainous part of the state was surveyed by one set of crews, while the eatern, flatter, part of the state was surveyed by a different set of crews.

Which set of crews do you think did a better survey, and why? (That is, which part of the state, western/rugged or eastern/flatish, was better surveyed?)

If you go to topozone.com and you start at the Missouri Base line, at the 5th Principal Meridian 36 30.00 the reading is near 91 00.30, at 37 00.00 the reading is 90 59.00 and so on, looks like it is a slightly curved line, I still dont understand if the 5th Principal is on or near 91 00 00 that the initial point has a value of 91 03 07, I sure hope that this is not racking your brains, I have heard all kinds of explanations for this but know exactly what was done and am trying to see if anyone else can figure it out as well. You do not have to have a license to be smart!!!!!!!!!!! only the will to want to know, maybe I will persue getting one. though!!!

If you go to topozone.com and you start at the Missouri Base line, at the 5th Principal Meridian 36 30.00 the reading is near 91 00.30, at 37 00.00 the reading is 90 59.00 and so on, looks like it is a slightly curved line, I still dont understand if the 5th Principal is on or near 91 00 00 that the initial point has a value of 91 03 07, I sure hope that this is not racking your brains, I have heard all kinds of explanations for this but know exactly what was done and am trying to see if anyone else can figure it out as well. You do not have to have a license to be smart!!!!!!!!!!! only the will to want to know, maybe I will persue getting one. though!!!

Marty

I only worked on one project in Montana, for a few months in 1984, so I cannot claim any expertise about the history of surveying there. I do understand your question though, and I would certainly have to guess that the plains were probably surveyed more efficiently than the mountains, but the contrary may be shown, particularly if the crewss in the mountains were more experienced and careful, and the crews on the plains were not. Perhaps Montana has a website where the public can pose questions and seek advice from land surveyors, as some states do, and you can inquire that way.

Here is a table of "feet per 0.001 minute (miliminute)" that should cover the entire US range of latitudes (and I suppose much of Europe and the southern hemisphere as well). This too I logged somewhere in July 2001. I hope the format takes. (It didn't. I had to edit it.) It is based on: feet/mmin = 6.074 * cos(latitude).

Find the row which is the degrees of your latitude and the column which is the minutes of your latitude. The entry there is the ft/mmin of longitude at that latitude. The ft/mmin of latitude is always 6.074 ft/mmin, for any longitude.

Feet per 0.001 Minute of Longitude at the Degrees and Minutes of Latitude Tabulated

____min __0___5___10___15__20__25__30__35___40__45__50__55__

deg

26______5.46 5.46 5.45 5.45 5.44 5.44 5.44 5.43 5.43 5.42 5.42 5.42

27______5.41 5.41 5.40 5.40 5.40 5.39 5.39 5.38 5.38 5.38 5.37 5.37

28______5.36 5.36 5.35 5.35 5.35 5.34 5.34 5.33 5.33 5.33 5.32 5.32

29______5.31 5.31 5.30 5.30 5.30 5.29 5.29 5.28 5.28 5.27 5.27 5.26

30______5.26 5.26 5.25 5.25 5.24 5.24 5.23 5.23 5.22 5.22 5.22 5.21

31______5.21 5.20 5.20 5.19 5.19 5.18 5.18 5.17 5.17 5.17 5.16 5.16

32______5.15 5.15 5.14 5.14 5.13 5.13 5.12 5.12 5.11 5.11 5.10 5.10

33______5.09 5.09 5.08 5.08 5.07 5.07 5.07 5.06 5.06 5.05 5.05 5.04

34______5.04 5.03 5.03 5.02 5.02 5.01 5.01 5.00 5.00 4.99 4.99 4.98

35______4.98 4.97 4.97 4.96 4.96 4.95 4.94 4.94 4.93 4.93 4.92 4.92

36______4.91 4.91 4.90 4.90 4.89 4.89 4.88 4.88 4.87 4.87 4.86 4.86

37______4.85 4.85 4.84 4.83 4.83 4.82 4.82 4.81 4.81 4.80 4.80 4.79

38______4.79 4.78 4.78 4.77 4.76 4.76 4.75 4.75 4.74 4.74 4.73 4.73

39______4.72 4.71 4.71 4.70 4.70 4.69 4.69 4.68 4.68 4.67 4.66 4.66

40______4.65 4.65 4.64 4.64 4.63 4.62 4.62 4.61 4.61 4.60 4.60 4.59

41______4.58 4.58 4.57 4.57 4.56 4.56 4.55 4.54 4.54 4.53 4.53 4.52

42______4.51 4.51 4.50 4.50 4.49 4.48 4.48 4.47 4.47 4.46 4.45 4.45

43______4.44 4.44 4.43 4.42 4.42 4.41 4.41 4.40 4.39 4.39 4.38 4.38

44______4.37 4.36 4.36 4.35 4.34 4.34 4.33 4.33 4.32 4.31 4.31 4.30

45______4.29 4.29 4.28 4.28 4.27 4.26 4.26 4.25 4.24 4.24 4.23 4.23

46______4.22 4.21 4.21 4.20 4.19 4.19 4.18 4.17 4.17 4.16 4.16 4.15

47______4.14 4.14 4.13 4.12 4.12 4.11 4.10 4.10 4.09 4.08 4.08 4.07

48______4.06 4.06 4.05 4.04 4.04 4.03 4.02 4.02 4.01 4.00 4.00 3.99

49______3.98 3.98 3.97 3.96 3.96 3.95 3.94 3.94 3.93 3.92 3.92 3.91

[This message was edited by Don&Betty on August 15, 2002 at 06:46 PM.]

[This message was edited by Don&Betty on August 15, 2002 at 06:50 PM.]

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[This message was edited by Don&Betty on August 15, 2002 at 07:10 PM.]

[This message was edited by Don&Betty on August 15, 2002 at 07:18 PM.]

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