Recently I was asked by a local Rav in my hometown of Manchester to carry out a small calculation. He was planning to build a new Shul and wanted the (so called) Mizrach wall to precisely face the right direction, that is towards Jerusalem.
Well, I said to myself, this is a simple fifteen-minute job which can be accomplished in three easy steps. Furthermore, it will give me the satisfaction of having contributed to a Hiddur Mitzvoh. Firstly, get out the local Ordnance Survey map and, with a protractor, measure the exact angle between the Shul street and North. Secondly, open my atlas and similarly measure the angular degrees between the North, Manchester and Jerusalem. Thirdly, mark the resulting line on a local street map for handing over to the architect. (In passing, I found it of interest that the Oruch Hashulchan - reference later - refers to planning restrictions regarding the orientation of buildings in his home country of Lithuania.)
Step one was easily done. But when I came to step two that I realised that it was not going to be simple. What I discovered was that there is more than one way to measure the direction of Jerusalem from Manchester and, indeed, from any place where Jerusalem is below the horizon. Moreover, the Halachically correct way seems to be in dispute amongst several Achronim including the Levush, Emunas Chachomim, Yaavetz and Tanya.
This is not meant to be a Halachic article as such, but it is clearly agreed amongst the Poskim (O.C. 94) that one should stand facing Eretz Yisroel in Tefilloh. (There are different views in some Rishonim; see a summary in the Oruch Hashulchan (loc. cit), but this is the majority and accepted view). Incidentally, if you have a "Mizrach" plaque displayed on your dining-room wall it is almost certainly a double misnomer: (a) the required direction is probably not East and (b) one should not face directly East in Tefilloh (Remo 94:2).
By now you might be wondering, "What is the problem? Can't one just face Eretz Yisroel?" To explain this issue a modicum of mathematics and some facility for visualisation are needed. In the hope of encouraging a high proportion of readers to stay with me to the end, I shall try to keep the former to a minimum and assume that everyone knows what an angle is, what a right-angle is and how angles are measured in degrees, with 360 degrees indicating the starting point.
It is an axiom of geometry that the shortest distance between two points is a straight line and that there is only one such line between any two points. Since the Earth is (nearly) spherical the only straight line between a mispallel and Jerusalem is a tunnel through the Earth - imagine a long needle piercing a grapefruit.
How does one translate or adapt this fact to the requirement of Chazal? Take this same grapefruit and mark two dots on its surface some distance apart, one dot representing Jerusalem and the other your home-town, say London or New York. You will find that you can join the two dots by an infinite number of 'straight' or even (unwobbly) curves on the surface. All these curves are routes or directions from one dot to the other. More careful observation (try measuring with string or elastic) will show that there is one curve which is shorter than all the rest. This curve, if continued round the sphere, is known as a Great Circle.
You can discover the most important properties of this curve by carefully slicing the grapefruit in two along this last line, ensuring that you have a straight cut. The cut will go through the centre of the fruit and will create two equal halves called hemispheres. Cutting along any of the other lines joining the two dots will result in an unequal division. There are, of course, many ways of cutting a grapefruit which give two equal halves, all such cuts being along Great Circles. But given two specific points on the surface (not precisely opposite each other), there is only one Great Circle which passes through both. Before eating this sliced grapefruit one further relevant point should be noted. This exercise with our model globe was done without recourse to marking poles and lines of Longitude or Latitude. The importance of this point will become evident later.
Now if you don't have a geographical globe to hand and don't feel like fiddling around with another grapefruit, find an atlas which includes maps of the whole Earth. In truth you will now be holding a book of lies! Since the Earth is spherical any flat representation must involve stretching, compression and distortion. Imagine a globe made of sheet rubber. When you try to place it on a flat surface, distortion is inevitable. When we look at a map of a large area, we need a sense of relative position, relative size, relative direction and relative distance. No single flat map can supply all of these requirements. Mapmakers use certain techniques, known as projections, in order to represent all or part of the globe's curved surface on flat paper. We will come to projections later.
Looking at your globe or world map (which may be displayed in rectangular form or as two circles representing hemispheres) you will see the North and South Poles (usually at the top and bottom) and lines of Latitude and Longitude, stretching across and down. All lines of Longitude pass through both poles and all are Great Circles, dividing the Earth into two equally sized hemispheres. At right angles to these Longitude lines are the lines of Latitude. One line of Latitude, half way between the North and South Pole, is known as the Equator; the rest run parallel to the Equator to its north and to its south. For convenience the Equator is designated as zero degrees and all other latitudes are referred to in terms of North and South (of it), with the North Pole being 90 degrees North and the South Pole 90 degrees South. England, for example stretches from 50 to 56 degrees North and Argentina from 22 to 55 degrees South.
Unlike Latitude, there is no natural starting point for Longitude. Several differing frameworks have been in use over the ages. After much discussion and debate, with the French being the last to fall in line, London, i.e. a marked line in the old Greenwich observatory, was accepted internationally as the zero Longitude. All other Longitudes are now referred to as degrees West and East (or plus and minus) from Greenwich.
Now (and this is the critical point) there is only one line of Latitude which is a Great Circle - the Equator. All other Latitudes are Small Circles; a slice through the Earth along them is like cutting the top off an egg. Such a slice does not go through the Earth’s centre and divides the globe into unequal sections. Since Latitudes are not Great Circles, it follows that the shortest distance between two points on the same Latitude does not follow the line of the Latitude. Moscow and Glasgow (both around 56 North) are nearly the same distance from the Equator, but a pilot following the Latitude line will add many miles to his journey; a shorter route (in the Northern hemisphere) will veer first to the North and curve later to the South. Near the North Pole the shortest route is often over the Pole as airlines are aware.
Depending upon the map's intended purpose differing projections (i.e. techniques for showing solid objects on flat surfaces), each with its own compromises, are used for individual maps. For example, one projection gets relative areas right, but fudges on direction, another just the opposite. One of the best known projections is that of Mercator, a 17th Century Flemish geographer. Mercator Projections distort areas (a Mercator world map shows Greenland much larger than Brazil; in fact, Brazil is nearly four times the size of Greenland!), but possess one quality which was of inestimable use to navigators before the modern age. Any straight line drawn on a Mercator map will always show the same angle in relation to the Latitude and Longitude lines. A sailor in possession of a compass and a Mercator chart could be confident that as long as he stayed with the same correct compass bearing, (e.g. 22 Degrees South of East), he would get to his destination. Such a line is known as a Rhumb line or a loxodromic curve. I will spare you the mathematics, but it can be shown that since a Rhumb line uses Latitude lines as well as Longitude lines, and the former are not Great Circles, a journey between two points following a Rhumb line will always be longer than the same journey following a Great Circle line. The exceptions to this are when the Rhumb line route coincides with a Great Circle. This only happens when both points are on the same Longitude line or are both on the Equator.
If you remember the demonstration with the grapefruit, in order to identify a Great Circle route, one does not need the conventions of North and South Poles, nor the north, south, east and west directions which are convenient usages thrust upon us by the Earth's daily rotation on its axis. Were this axis of rotation to be altered (say through a large asteroid hitting the Earth), North, South and the seasons would change; the Rhumb line route between London and Jerusalem would follow a different path, but the Great Circle route between any two locations would remain the same.
Imagine a vertical flagpole (one that points to the centre of the Earth) being erected in Jerusalem. If extended sufficiently high it would theoretically be visible from half the globe (over 730 miles tall for London and 26,000 for New York). Somebody facing this flagpole during Tefilloh can surely be described as facing Jerusalem. The line of his sight would be following the Great Circle direction, not that of the Rhumb line, as we can now demonstrate. Visualise a perfectly flat massive rectangular board. One edge of this board is attached vertically to the Jerusalem flagpole like an estate agents ‘For Sale’ sign.. The opposite edge should be visualised as reaching the distant Mispallel., so that the sides of the board follow the direction of the Tefilloh. Since the flagpole points to the centre of the Earth, this board, being attached to it, would reach the centre of the globe, if extended in the same plane. Any such board which goes through the globe’s centre must divide the Earth into two equal parts. As mentioned earlier, such a division defines a Great Circle on the Earth’s surface.
To summarise, amongst the many possible routes between two points on Earth, or for ‘Mizrach’ from your hometown to Jerusalem, there are two likely candidates. They are the shorter Great Circle route with its continuous change of compass bearing or the longer but simpler Rhumb line with its constant compass bearing, the latter being derived from the Earth's daily rotation on its axis, (Ketzoroh Ve'arucho and Arucho Ve'ketzoroh).
Let us now turn to the Halachic implications of the above. Chazal instruct us to face Jerusalem during Tefilloh . Did they mean a Great Circle line, or a Rhumb line or something else? One should bear in mind that the difference between the first two directions is very small in countries near to Eretz Yisroel, where nearly all Yieden lived in the times of Chazal. (Why Chazal referred to Bovel as being North of Eretz Yisroel is a related but separate issue). Even in England (see appendix) the difference between the two directions is comparatively small. Let us say you want to build a ‘Mizrach’ wall 40 foot long and have decided where one end should be placed. You now have to mark the other end of the ‘Mizrach’ and so fix the line of the wall. In London, choosing between a Rhumb line orientation and a Great Circle one means moving the second end about 9 ft. But, in San Francisco, the second end for such a wall would be around 48 ft.
There is little guidance to be found in the Rishonim on this issue. Attempts to derive the Rambam's view from his statements (see Yad H. Beis Habechirah 7:9; Perush on Mishnah B.Basro 2:9; attributed commentary to Rosh Hashonoh, quoted by the commentary on Kiddush Hachodeh 7:1) appear far-fetched to this writer. Turning to the Acharonim, one may first mention that the Perisha (94:1) (17th Century), discussing the direction of Tefiloh, uses the analogy of an arrow being shot from a bow. We do not know whether he was aware of our problem, but long distance missiles in their parabolic curves follow a Great Circle path. The first Acharon to talk at length about 'Mizrach' orientation is a contemporary of the Perisha, the Levush (94). Besides his greatness in Torah, he was an expert in mathematics and astronomy. Taken at face value, his text supports the Rhumb line route.
This reading of the Levush is certainly the understanding of Rav Aviad Sar Sholom in his profound work, Emunas Chachomim. (Since Rabbi Aviad is not that well known one may mention that the opus of this 18th Century Italian Talmid Chochom of Mantua attracted lengthy footnotes from R' Yaakov Emden in the latter's work, Mitpachas Seforim). Rabbi Aviad disagrees with the Levush and in forceful language opines that the correct line is the Great Circle one. Rabbi Aviad's view is supported by R' Yisroel Segal of Zamocz in his work, Netzach Yisroel (published in 1741). R' Yaakov Emden in Moir U'Ktzioh (Ch: 150) quotes, without further comment, R' Aviad's animadversions on the Levush. Moving on about a century, the Rav (Ba’al HaTanya) in his Shulchan Oruch deals extensively with the astronomical aspects of "Mizrach". He does not refer to our problem explicitly, but his approach is in congruence with the Great Circle reckoning.
After a period of quiescence this debate has been revived in recent years in Torah journals and this writer wishes to acknowledge that his research has been facilitated by others' efforts. Both the Rhumb line view and the Great Circle view have their protagonists. Amongst contemporary Talmidei Chachomim, R' Yechiel Silber of Bnei Braq (in his Birur Halochoh - Tinyono and Chamishi, both O.C. 94) pens an extensive treatment of the problem and clearly favours the Great Circle route.
An appendix lists both possible directions for several cities. All angles are measured clockwise from true North, e.g. since East is 90 degrees, 124 clockwise (deducting the 90) is equivalent to 34 degrees South of East - a little over one third of the angle going from East to South. When using a magnetic compass, note that the Magnetic (compass) North is currently around 4 degrees West of true North in the U.K. and an allowance for this should be made.
Even if the Great Circle line is to be preferred, there may be a Halachic problem of adopting it in towns where the Minhag is otherwise. That is an issue which I am happy to leave to the Baalei Halochoh.