Coordinata Magica (Magical Coordinates) derive from the mystical insights of the great Ptolemy, who delineated a system of coordinates in his Geographica. A magus who successfully completes a sequence of research based on Geographica may use a new Hermetic Range: Coordinatum, which is equivalent to Arcane Connection (+4 magnitudes). Coordinata Magica are based on the Law of Names, rather than the Law of Contagion, and represent a means of identifying any point on the surface of the earth. Points are relevant to within about five hundred paces, similar to the variance for the use of astrological time (TMRE, 49). More accurate results are extremely difficult using astronomical readings, even with the aid of Hermetic magic, due to the Limit of Nature (the nature of the stars themselves, TMRE, 49).

Determining coordinates relies on a methodology of measuring the array of fixed stars through astronomical observation, which is detailed in Geographia. Astronomical observation requires an astrolabe or similar tools and a clear view of the stars for four hours. This may be shorted to two hours if the observer has access to accurate magical measurement of time, such as that provided by Sense the Hour (TMRE, 51). Following measurements of the stars, about four hours must be spent on calculations, which may be shortened to two hours if the 'handy tables' from Geographia are available. Thus, without a magical means of reckoning time or the 'handy tables,' a measurement of the array of fixed stars requires about eight hours, at least four of which must be under a clear night sky.

An accurate copy of the handy tables from Geographia adds +1 to attempts to measure the array of fixed stars, in addition to shortening the time required for calculations. If measurements are made on the night of a solstice or equinox, an additional +1 situational bonus is applicable (although calculations may be made at a later date without affecting the bonus).

Ptolemy's system of coordinates relies on an Origo, or zero point, which is computed from a measurement of the array of fixed stars at a far western location. By comparing measurements of the array of fixed stars from two different locations, the difference between the two measurements may be reduced to two numbers. One number represents a measurement of the north-south difference, and the other the east-west difference (similar to modern longitude and latitude measurements, although with a different zero point). If the two locations are distant enough from one another, then a meaningful measurement of difference can be computed. Both north-south and east-west differences must be computed very accurately (each results in a corresponding Insight, see below).

Insight One: Coordinata Magica represent a mystical reality, which may be perceived by an appropriate Intellego Vim effect. This Insight may be gained by learning the method described by Ptolemy in Geographia and measuring the array of fixed stars, and is completed by inventing the following spell:

Insight Two: Coordinata Magica require an accurate measurement of the size of the Earth, which must be confirmed by measuring the array of fixed stars and casting Percipio Locus Mysticus for two locations which are widely separated north and south (such as Caledonia and Africa); the more widely separated the two locations are, the more accuracy will be possible for this Insight. This insight is completed by making the necessary measurements along with casting the spell, and then by inventing the following spell:

Insight Three: Coordinata Magica require an accurate calculation of the Origo, based upon measurement of the array of fixed stars and use of Comparo Ordines Stellarum Fixarum for locations widely separated both east and west but on roughly the same latitude (such as Thule and Rus, or the Fortunate Islands and southern Syria); the western location must be farther west than mainland Mythic Europe. This insight is completed by making the necessary measurements and comparison using Comparo Ordines Stellarum Fixarum, and then by inventing the following spell by experimentation, an improved version of the previous spell that now uses a new Range:

Note that when making comparison measurements, the comparison at the first location (east or west) must be made in comparison to an existing, known location, for which the array of fixed stars has been measured and perceived with Percipio Locus Mysticus. The comparison at the second location must compare the first location with the other two points. Thus, Comparo Ordines Stellarum Fixarum must be cast three times in total.

Insight Four: Coordinata Magica may be substituted for Arcane Connections to points on the surface of the earth (anywhere from which the night sky is visible; although clouds do not prevent such effects from functioning). It is not necessary to include a Vim component to a spell which utilizes the new Range, and in fact Coordinata Magica could be subtituted for Arcane Connections in some relevant guidelines. This insight is completed by inventing the following spell:

After completing Insight Four, the researcher will have accumulated twenty-five magnitudes of Breakthrough Points, and will be able to create other spells using the new Range:

The researcher will need to create another spell using the new range to complete the Breakthrough. The new spell may be any spell that incorporates Coordinata Magica, but should be fifth magnitude or greater, in order to complete thirty magnitudes of Breakthrough. Following the invention of this spell, the researcher may author a tractatus on Hermetic Theory that describes the Breakthrough; at least one lab text should be included for a spell that demonstrates the new Range. The researcher may also author a tractatus on Artes Liberales that describes the method and defines the correct Origo for useful measurements of the fixed stars (without knowing the correct Origo, mystically useful Coordinata Magica cannot be determined, although they can be used).

Note that it is possible to design a spell that incorporates a specific Coordinatum Magicum, which may not be varied. Such a spell might allow someone to scry on a specific location, for example, but no other. Other spells may be designed to use any Coordinatum Magicum. Note that for each Coordinatum Magicum, there is a unique corresponding place on the surface of the earth (within 500 paces); if a Coordinatum Magicum is incorrect, the spell might still function, but at the location that correctly corresponds to the Coordinatum Magicum used.

Because of issues of measurement accuracy and variance (see below), and the limit that the night sky must be visible from the location of a Coordinatum Magicum, large open fields or similar areas are most suited for this form of travel. For flying magi, arriving at a height above the ground may be sufficient, but because of the inherent variance possible, even that approach is no guarantee of success, and arriving more than 500 paces above the ground would be invalid for any set of coordinates that were not made at that height (and probably much less, given variance). Similarly, coordinata magica for locations of great altitude, if coordinata magica are not specifically measured at that location, might manifest too much variance to be valid. Variance may be estimated in advance by casting a spell such as Speculor Coordinato Magico immediately before leaping.

Because Coordinata Magica are only inherently accurate to within 500 paces, the accuracy of the measurement of the array of fixed stars can influence the amount of variance experienced in use of a given Coordinatum Magicum. Measurement of the array of fixed stars can vary with time and season, even if measurements are made with perfect accuracy.

For each point on the surface of the earth, there is a corresponding Coordinatum Magicum; however, a given Coordinatum Magicum corresponds to an area about 1000 paces in diameter (an area roughly equal to one hundred standard Boundaries). Three terms are relevant to discussion of accuracy and variance:

Punctum Mensurae (Point of Measurement) is the location from which the array of fixed stars was measured; variance is determined from this point

Punctum Variationis (Point of Divergence) is the location within the corresponding area (about 1000 paces in diameter) where use of Coordinatum Magicum in a magical effect will be centered

Punctum Finis (Point of Purpose) is the location, after adjustment through use of Finesse, at which a Coordinatum Magicum effect will actuate

One may reduce the potential variance somewhat, through more accurate measurements of the array of fixed stars. Basic success must reach an Ease Factor of 15, as noted above, and this guarantees measurement valid to within 500 paces. I.e. Punctum Variationis will be within 500 paces of Punctum Mensurae.

For every three points by which that Ease Factor is exceeded (i.e. 18, 21, 24, etc.), the maximum potential variance is reduced by about 100 paces (from 500 paces). Thus, if the measurement roll was 24+, variance could be reduced to no more than 200 paces at most, and higher rolls could eliminate potential variance entirely. I.e. in a perfectly accurate result, the caster would arrive exactly at the point where the measurements were made. However, this would require a measurement of the array of fixed stars 30+; although possible, such a result is highly unlikely, even for a superb astronomer with superb measurement instruments and a perfect view of the night sky, and depends as much on dumb luck (die roll) as skill.

Repeated measurements of the array of fixed stars at the same exact location may gradually accumulate accuracy; treat the number of repeated measurements (which must be at least one week apart from any previous measurement throughout the year) as experience points applied to an Ability, and use the corresponding "Score" as a bonus to the next measurement. Because a year contains only 52 weeks, the total bonus from repeated measurement cannot exceed +4.

Through use of Finesse, a caster may "offset" Punctum Finis from Punctum Variationis. By exceeding the relevant Ease Factor (which may differ depending on the specific effect in question), the caster may offset Punctum Finis from Punctum Variationis:

This is most useful when Punctum Variationis has been recently estimated through use of a spell such as Speculor Coordinato Magico. 200 paces is the maximum possible adjustment of Punctum Finis away from Punctum Variationis.

Note: although this material is loosely adapted from Ancient Magic (The Hesperides), it does not conform to the published material. Notably, although Ancient Magic states that coordinates "will not revolutionize the Order," it then proceeds to later list a series of consequences that are demonstrably significant, with great potential to "revolutionize" the Order. For that reason, among others, these rules require a process of Integration similar to that for other topics in Ancient Magic.

A second error is made in stating that only someone with the Celestial Magic Mystery can determine the array of fixed stars through a spell. The guideline referenced by the sample spell in Ancient Magic (p91) is available to all magi, as stated in TMRE (p51). An additional unforced error was made in stating that the Celestial Magic Mystery allows a magus "to cast spells which affect starlight." There is nothing in TMRE that indicates anything remotely resembling this statement, and it is unnecessary for the guideline referenced.

The statement (p91) that "the size of the Earth is known" is misleading. The latter portion of Ptolemy's Geographia, containing an enormous listing of various locations with their coordinates, is wrong. This could actually have been known at the time it was authored, as Ptolemy used an estimate of the circumference of the earth that was too small (one similar to that of Posidonius, rather than the older and more accurate estimation of Eratosthenes). Ptolemy had no way of knowing that the newer estimate was less accurate, and may have had reasons to believe it was more accurate: hence his use of the smaller estimate.

As a result of this estimation error, and also due to Ptolemy's use of many traveler's estimates of the distance between locations, many of the coordinates he listed are wrong. This is especially true for locations east of Mythic Europe, or north of the Mediterranean lands. Thus, the listed coordinates in the latter portion of Geographia are of dubious value, and in fact may mislead readers by causing them to use the same erroneous measurements as Ptolemy. The system of measurement using the array of fixed stars is not flawed in itself, however, and can yield accurate coordinates.

Because many of Ptolemy's coordinates are in error, particularly those far distant from Alexandria, the Origo (zero point) is in fact also in error, and must be re-calculated based upon a measurement taken at a far western point, such as in the Fortunate (Canary) Islands or, alternatively, Thule. Thule actually represents a potentially superior location for taking such measurements, as the western coast of Thule is about two hundred miles farther west than the westernmost Canary isles.

Once several accurate measurements have been taken at such a far western location, the correct Origo for Coordinata Magica may be calculated. This represents the third Insight referenced above, and is required for the Breakthrough. For the Bohemian Rhapsody Saga, neither the Fortunate Islands nor Thule lie on the longitude of the correct, mystical Origo. That point lies on the equator, and on the longitude of the western side of the isle of Flores in the Azores (at approx. 31 degrees, 16 minutes West using modern coordinates). This does not have to be measured directly; it is sufficient to measure accurately at a point far enough west that the inaccuracy of Ptolemy's coordinates can be both demonstrated and corrected for (someplace more than 16 degrees West in modern coordinates).

NOTE: once the correct Origo has been determined, all measurements of the array of fixed stars must be re-calculated based on the new Origo to be valid as coordinata magica. Previous calculations, which used the point designated by Ptolemy for his system, are invalid for further use. If the original measurements were recorded, it may be possible to re-calculate, but this requires an Int + Artes Liberales roll 18+; it may be necessary to re-measure and calculate coordinata magica based on the new Origo.

It is also worth noting that it is not possible for the correct Origo to be determined without access to the perception of mystical reality bestowed by the Insights & effects listed above: mundane calculations may be sufficient to demonstrate the errors in Geographia, but are incapable of determining where the correct, mystical, Origo must be.

Finally, although the location of the mythical Hesperides might be the Canary Islands, there are other possible locations, and if the garden is in fact a regio, then it might have multiple possible entrances. For this reason, the Bohemian Rhapsody version of the Canary Islands does not correspond to the version published in Ancient Magic. While some material is used, some is not.

Further, it is not at all clear that the "Purple Islands" of antiquity correspond with the Canary Islands; in fact, Pliny the Elder specifically states otherwise, placing the Purple Islands at some distance from the "Fortunate Islands," which are described as corresponding to the Canary Islands (by naming the constituant islands). Pliny's description of their relative location is confusing, but one possible location for Pliny's Purple Islands is actually Mogador, where purple dye was obtained in antiquity. The Canary (Fortunate) Islands lie just over 250 miles south and west from there, which would roughly fit Pliny's description. Another possiblity is Madeira. In either case, Pliny was clearly of the opinion that the Fortunate Islands are a separate place from the Purple Islands.