THE MAP of Washington, the story goes, encodes certain secret symbols of Freemasonry, symbols that go back centuries if not entire millenia. Symbols, moreover, that if properly decoded will reveal a cryptic history not only of the capital but of the country. And not only a hidden national history, but a hidden destiny as well.

That there is something suspicious, or at least suggestive, about the city's geometry has been rumored for a very long time, no doubt since the heyday of anti-Masonic paranoia during Andrew Jackson's administration. Now, with the explosion of popular interest in conspiracies, and the occult histories they necessarily imply, such claims are again growing in number and volume.

Indeed, an apotheosis in this genre was reached at October's Million Man March, when Louis Farrakhan, standing on the Capitol's West Front, told a baffled national audience that the Capitol, the Washington Monument, the White House and the whole of the city's monumental core had been "placed and based in a secret Masonic ritual." Farrakhan went on to sprinkle his extended remarks liberally with references to Masonic initiation rites, numerology, Egyptology and other arcana.

When it comes to the national capital's map, however, and the patterns that may have been traced into it by Freemasons, there are other thinkers whose visions either complement or compete with Farrakhan's, and in the interest of context, these too bear examination.

A word or two of initiation. Masons have a confused place in the national imagination: To many they are charitable, fun-loving lodge-members who engage in odd but harmless fraternal rites; to some, their secrecy is suspicious; to a few they are Satanists working to destroy Christianity and take over the world; a handful of nonmembers identify with their precepts. Although many of the nation's founders were Masons, including George Washington, Ben Franklin, Paul Revere and all the participants in the Boston Tea Party, there has been a persistent conspiracist underground that for 200 years has regarded them as a front for some other force -- the Bavarian Illuminati, the International Jewish Conspiracy, pagan Devil-worshippers of Atlantis, etc. -- which seeks to undermine American institutions and sovereignty.

Currently, there is a large and ever-increasing library of new anti-Masonic literature that associates them with the godless New World Order that purportedly threatens the country's independence and its citizens' freedoms; the evidence that Washington, D.C. is now and has ever been in the grip of this conspiracy is, according to these authors, plentiful.

This includes above all the particulars of the Great Seal pictured on the back of the dollar bill, with its undeniably Masonic symbolism of an unfinished pyramid, the All-Seeing Eye and the Latin motto that translates into "New Order of the Ages." Even the number of feathers on the eagle of the Seal's obverse has been linked to the number of degrees in the Scottish Rite of Freemasonry. That the Seal appeared on the currency during the presidency of Franklin Roosevelt, a 32nd degree Mason and the purported prime mover of socialism in America, is said to be no coincidence.

Many details of the city are cited to demonstrate an ages-old, un-American, un-Christian allegiance of the national leadership. That the Statue of Freedom atop the Capitol faces not west toward the rest of the country, but east, toward Egypt and Persia, is one such sign. That the monument to George Washington is an Egyptian obelisk is another. That the cornerstones of many government buildings, including the Capitol, were laid by Masons in full-aproned regalia, performing mystifying Masonic rituals, is yet another. Clinching proof of conspiracy is that the patterns of the capi tal's very streets conceal Masonic shapes, symbols and signsfrom all but a knowing, if not all-seeing eye.

The simplest version of the Masonic map theory is the claim that the Mall is laid out so as to replicate a Masonic Lodge, and a glance at a typical floor plan of an 18th century Lodge quickly suggests where the idea could have come from: There is an apparent affinity of form. According to Masonic histories, the shape of a Lodge had been largely standardized by the time Washington was laid out in the 1790s; it consisted of a long rectangle positioned east and west, with a tapered but unfinished triangle at the eastern end, where the "furniture" (compass, square and Bible) would be found, and where the Master would officiate. This does not necessarily describe a building, by the way; lines on a floor would also constitute a Lodge.

That's essentially the Mall. It's a long east-west rectangle with an unfinished triangle at its east end, formed by the convergence of Pennsylvania and Maryland Avenues at the foot of Capitol Hill. Where the Master would stand, stands the Capitol. Certain details are absent, but other important Masonic shapes can be located on the Mall. An aerial view of the Washington Monument and its circular walkway, for example, provides the "Point in the Circle," a multi-layered symbol in Freemasonry's speculative and mystical side.

Of course, circles, squares, rectangles and triangles are not hard to find on any city map, especially Washington's with its Baroque-inspired design filled with circles and radiating boulevards. But the presence of these shapes at the very heart of power is not coincidental to a suspicious observer, especially one familiar with the importance Masons place in shapes, and the symbolic significance they have attributed to them. Masons maintain that though their Lodges can be dated only from 1717, Freemasonry actually developed out of medieval builders' guilds. Accordingly, they are thus keeping alive a tradition that is not only responsible for the sacred architecture of Gothic cathedrals, but also an inner doctrine of symbolic geometry that they say dates all the way back to the construction of Solomon's Temple. This doctrine regards the character of God as that of the Great Architect, and of builders as potentially doing God's work, though that would depend on the purpose, shape, proportion and harmony of their creations.

This is an abstruse doctrine, but one with a long history. Its origins lie with the mystical Pythagoreans of ancient Greece, who believed all of creation was continuous and could be expressed in numbers; musical octaves, for example, are produced by striking strings whose lengths are in whole number ratios. Eventually, the Pythagoreans were confronted with heretical irrational numbers, which they apparently attempted to suppress.

Related doctrines were at large in Medieval times. The Benedictines, for example, are sometimes credited not only with building their monasteries according to sacred geometric principles, but even of pitching their chants in harmony with their architecture. Gothic architecture is of course replete with complex geometrical symbolism. Bernard of Clairvaux is said to have expressed its underlying concept in the 12th century when he asked, "What is God?" and answered, "He is length, width, height and depth."

Renaissance hermeticism was obsessed with grasping the harmony of creation's ultimate continuity; the occult means it used for the purpose -- sacred geometry, numerology, Kabbalah, alchemy, magic, etc. -- became, as scholar Frances Yates has demonstrated, the foundation of the scientific revolution. But the point of this ages-old search was to apply the underlying mechanics of cosmic harmony to man and his worldly works.

Thus, a building designed according to such principles is not merely a building; it is an inspired and in a sense, a living thing, resonating with the cosmos. And if that could be true for a building, it could be true for the ideal city, if only such a thing could be designed and built. Which brings us back to Washington and the Mysteries of its design.

The obsession with geometrical forms to be found in the works of speculative Freemasonry is an echo of this tradition, though no one knows how it came to be part of Masonic doctrine; Freemasonry's ultimate origins are a matter of complete confusion, with at least a dozen competing theories in circulation. Still, if one concludes that there is a Lodge traced into the Mall, it is not merely a matter of a floor plan: Whether sacred or profane, it would be a Lodge.

But Lodges are not all that interpreters of the city's map have discovered there: There may be two Ideal Cities traceable within the Profane City's limits.

In 1989, Michael Baigent and Richard Leigh published a work entitled, "The Temple and the Lodge," which sought to demonstrate that Freemasonry evolved from the 14th century wreckage of the Knights Templar, a Crusader group of immense wealth that was accused of heinous blasphemy and suppressed. Their concluding chapter is a Masonic history of the American Revolution, and ends with the design of Washington, D.C. According to them, George Washington and Thomas Jefferson intervened in Pierre L'Enfant's work of laying out the city, so as to impose a pair of octagonal shapes around the Capitol and the White House.

Baigent and Leigh, who cite no sources for their assertion, write that the intervention of Washington and Jefferson would "produce specifically octagonal patterns incorporating the particular cross used as a device by Masonic Templars."

Actually, it's well-known that Jefferson, who is not known to have been a Mason, wanted the street pattern of the federal city to be a simple grid; he had produced a grid layout of the city before L'Enfant was assigned the task and complained to the Frenchman about all the diagonal avenues he was planning.

Anyway, it was not octagons that interested 18th century revolutionaries; it was circles and triangles. James H. Billington, writing in "Fire in the Minds of Men," describes at length the connection between the concepts of Pythagorean order and of revolution that developed during the period. "However bizarre it may appear to later revolutionaries and historians alike," writes Billington, "this Pythagorean passion seriously influenced the organizational activities of the first revolutionaries."

According to Billington, revolutionaries of the period combined their political thinking with geometry; odd though it may seem, it was a case of Pythagorean simplicity warring against Rococo absolutism. Revolutionaries signed their names with geometric shapes. Ruminations about the social and political implications of the circle and triangle may be found in the period's revolutionary works, including that of the notorious Adam Weishaupt, founder of the very Illuminism that still figures in the nightmares of modern conspiracists.

This state of mind may appear incomprehensible today, though it is worth remembering that Isaac Newton had earlier established the foundation of modern scientific thinking at the very time he was trying to map the typography of Hell.

In any event, all of this political geometry is entirely germane to both the geometry-minded Masons, who were in the thick of the international revolutionary underground of the period, and to the Mysteries of the American capital's map, which is, after all, overflowing with both triangles and circles.

As for the octagons, if they are indeed carved into the city's street pattern, they would point not to a revolutionary ideal shape, but to a different but related ideal shape: The eight-sided Ideal City of Vitruvius, the 1st century B.C. architect whose work became an essential influence on early modern architects. In Vitruvius' time, the Ideal City could only be imagined; any effort actually to build one would necessarily fall short. His octagonal city, by the way, is based on the belief, current in antiquity, that there were eight winds. An effort to incorporate the concept in Washington's map would represent a notable effort of intellectual imagination.

Are these octagons there? No. At least not where Baigent and Leigh assert they are. They do provide a map purporting to show how the streets around the Capitol and White House encompass the shapes, but if the rule is that there actually must be streets where purported shapes are to be found, then their map falls short. Farrakhan, for his part, is interested in none of this; he's interested in Gizeh, Egypt, the site of the Great Pyramid and the Sphinx, which he believes that the nation's founders, as Masons, have purposely replicated in symbolic form in downtown Washington: The Jefferson Memorial is the Sphinx, Arlington Cemetery is the Valley of the Kings, etc.

Expansion on his references to the city's Mysteries can be found in the "Million Man March Home Study Guide Manual," published in advance of the march. There he provides a map of the Mall revealing the hidden pyramid that can be found by drawing lines connecting such major sites as the Capitol, the White House and the Lincoln Memorial, among others.

However, L'Enfant, who placed both the Capitol and the White House on their respective sites, is not likely to have in mind their relation to the sites of the later Lincoln or Jefferson Memorials, if for no other reason than that in the 1790s, when he was working, the sites of both memorials lay in the midst of the Potomac Mud Flats. A century was to pass before those locations were turned into dry land. Farrakhan, it should be noted, strongly approves of the secret Masonic designs he perceives in the city's map, because he believes the Masons to be serious adepts of Islam. That is also why derivations of the number 19, important in Islamic numerology, can be found all over the monumental core.

It is in fact possible, in some sense, to "Islamicize" the Masons if you connect them with the Templars, because the Templars are said to have maintained cordial relations with Muslims, especially the Ismailis, during their long Crusader sojourn in Jerusalem. Indeed, they are said to have been influenced by certain Islamic and Judaic doctrines, and to have evolved a mysterious Secret Doctrine, on which much ink has been spilled, based on those influences. Such scholars as Peter Partner dismiss this tradition as nonsense, but that is the folklore.

Farrakhan's view is not based on this connection, but rather on the fact that one kind of Mason, the Shriners, wears fezes and refers to "Allah" in their rites, rites which were created in the 19th century. The Home Study Guide Manual refers to all Masons as "Moslem Sons," notes that many presidents have been Masons, and states that "There are many others in high places who know and recognize the identity of Master Fard Muhammad {regarded as the promised Mahdi by the Nation of Islam}, the Honorable Elijah Muhammed and Minister Louis Farrakhan \. \. \. ."

Farrakhan's position that the Masons are Muslims is shared by the Free the Masons Ministries, located in Washington state, though they do not at all approve of this. To this Ministry, Islam is a form of paganism, the Masons are serving the devil, and the streets of Washington are the Masons' secret tribute to Satan. To demonstrate this, the Ministries published "Freemasonry: Satan's Door to America?" a pamphlet distributed in the late 1980s. Included in this work is the single most spectacular map ever drawn of the secret shapes discernible in the capital's streets. Not only is there the usual Mason's compass and square, there is an entire Pentangle traced out focusing on the White House, and a Satanic Horned Goat traced around the Capitol. Most impressive of all is that the shapes offered follow actual city streets, with a minimum of rounding off and cheating. This map is the work of a man named J. Edward Decker, who writes of these secret shapes that he is as much impressed "by the audacity of the thing as the conspiratorial nature of it." Decker notes that the inverted Pentangle is an instrument used to conjure evil spirits, and citing a colleague whom he identifies as a "former Mason and former Satanist," he argues that this design "is a spiritual door to the occult, a planned invasion by the powers of darkness," and that the nation has "been under siege from the first day our first president walked into the Oval Office."

When Jefferson warned L'Enfant that he was drawing too many diagonal avenues into the city's plan, diagonals that created all these angles and triangles, and that crossed in traffic circles, he likely did not have these problems of Masonic perception in mind; Jefferson believed right-angle grids were rational, and that L'Enfant's many diagonals were troublesome and arbitrary.

L'Enfant went ahead anyway, not because he was a Mason, but because, according to the classic account, he was much influenced by the Baroque layout of Versailles. His converging avenues have nothing to do with Pythagoras, and everything to do with the three-road patte d'oie, enabling a viewer to command an impressive vista. L'Enfant has been criticized for this, since the inspiration was absolutist, and its application to a democratic capital is questionable. As architect Paul Speiregen put it in a 1963 critique, "The whole plan of Versailles focused on the bedroom of Louis XIV. What could that kind of plan mean symbolically in the United States?"

But who sees Louis XIV at the convergence of Pennsylvania and Maryland Avenues? Almost nobody, not when there's so much to distract one: compasses and squares and Great Pyramids and Horned Heads of Satan and Ideal Cities where once there blew only the Potomac winds. All eight of them. Charles Paul Freund writes frequently for Outlook.