The Harmonic Unified Field Equations of Bruce Cathie:
A Critical Evaluation

For those who have read Bruce Cathie’s books there will be general agreement as to their basic format and composition. Each are found to contain an introductory set of chapters that lay out certain discoveries made by Cathie, which include primarily his discovery of an Earth energy grid matrix, and also a set of mathematical equations referred to by him as Harmonic Unified Equations, which he puts forward as an extension to the basic E = mc^2 energy equation in physics.

The first several pages in three of his most prominent books [1] are practically identical, with only minor differences. Once the basics have been covered, the bulk of the remaining chapters are concerned with certain discreet analyses of various phenomena in relation to the identified Earth energy grid, and also Cathie’s Unified Equations.

Like many no doubt who have purchased his books, this researcher has looked very carefully at his proposed Harmonic Unified Equations, and made a personal attempt to evaluate them, in light of the analysis presented by Cathie himself.

Now in terms of the equations, Bruce Cathie states in his books that they are based upon the Einstein energy equation; an expression of the total energy ‘stored’ in a given amount of matter:

Now the breakthrough that Bruce Cathie suggests is the solution to this equation, is to express M in terms of C, so that Energy is expressed purely in terms of the speed of light. Cathie’s substitution is as follows:

m = c + √(1 / c)

Consequently, with mass now given purely in terms of light, Cathie is able to substitute this new expression into the Einstein energy equation, to produce the following:

This newly derived equation constitutes one of a set of three equations discovered by Bruce Cathie, which he refers to generally as Harmonic Unified Field Equations in his books. The particular one as given here (above) he designates simply as Harmonic Equation 1. The other two are just slight variations upon this, though they all incorporate the basic equation for mass (M), expressed purely in terms of light speed.

Just how exactly then did Cathie arrive at his basic mass equation?

The mathematical values he identified that led him to the breakthrough are well noted in all of his books, and herein I reproduce them below, along with the labels Bruce Cathie himself assigns to them [2]:

                                              17025 Earth mass harmonic
                                              14389 Speed-of-light harmonic         
                                                2636 Unknown harmonic

On the Earth Mass Harmonic

The first of the three noted figures, Cathie explains, is what would be a 5 figure numeric value for the total volume of the Earth in terms of cubic nautical miles. With the correct number of 'zeros' given in the expression, this is equal to about 170250000000 cubic nautical miles, in full. In presenting his figure of 17025 though in the early chapters of all of his books, Cathie does not provide an exacting calculation with a full working out as to the total Earth volume. The '5 figure harmonic approximation' only is provided. However that may be though, a specific value is indeed given in a later section of one of his books; notably The Bridge to Infinity [3].

Here Cathie specifically states that ANY spherical type body will have a volume of 170180681200 cubic minutes of arc, and furthermore, that in the case of the Earth itself, 1 minute of arc (angle) is equal to 1 nautical mile. Importantly however, though this figure is oftentimes noted as a volume, it is also (especially as a numeric sequence) referred to many times by Cathie in The Bridge to Infinity as a “mass harmonic at the earth’s surface” (see pages 109 and 172). As to how Bruce Cathie actually derives the figure of 170180681200 cubic minutes of arc for the earth's volume, careful reconstruction is necessary to reveal the answer, for he does not explain this in his books:

1) The most advanced Earth model in use today is the WGS84 model, based upon an equatorial radius of 6378.137 kilometres, and an inverse flattening measure of 298.257223563. From this, one can further derive a value for the Earth polar radius, with both being converted into statute miles:

                      Equatorial radius = (6378.137 × 0.6213711922) = 3963.190591704 statute miles

                      Polar radius = 3963.190591704 – (3963.190591704 × (1/298.257223563))
                      = 3949.902764022 statute miles

2) From the above figures one may thus now proceed to determine a value for the nautical mile that would ‘accompany’ such an Earth:

                                    ((3963.190591704 + 3949.902764022) × PI × 5280) / 21600
                                    = 6076.819455305 feet

3) With the nautical mile established, one can now convert both the earth equatorial and polar radius values into this measure:

                   Equatorial radius: 3963.190591704 / (6076.819455305 / 5280) = 3443.519505245 nm
                           Polar radius: 3949.902764022 / (6076.819455305 / 5280) = 3431.974036324 nm

4) A certain pause is required here whilst one considers the next step. Ordinarily one would proceed, in light of the fact that the geometry of the Earth is that of an ellipsoid, to use the values for both the equatorial and polar radius of the planet, in the following formula, as is recognised for determining the volume of any given ellipsoid, of the same type as the earth:

                                                                     4/3 × PI × a × a × b

                                    Where:   a = Equatorial radius;      b = Polar radius

                                    Thus: 4/3 × PI × 3443.519505245 × 3443.519505245 × 3431.974036324
                                    = 170465971375.741 cubic nautical miles

5) Now as one can see then, the answer as calculated is not really close at all to that given by Bruce Cathie in his book The Bridge to Infinity: 170180681200. How can this be explained? Quite simply, Cathie does not actually use the recognised formula for determining the ellipsoid volume of the Earth. Rather, he makes use of a subtle modification of the basic formula. Instead of multiplying directly the equatorial radius, squared, by the polar radius i.e. a × a × b, he appears to first calculate an average radius of the two, and once generated, he then proceeds to ‘cube’ this value. Thus, the equation that he would appear to use to achieve 170180681200 is as follows:

6) From the above then, with just a slight discrepancy of some 15 cnm, it is very clear how Cathie derives his value for the volume of the Earth, 170180681200 cnm, as stated in his book The Bridge to Infinity. However, one can still see that there is a marked difference between a five figure harmonic of this value i.e. 17018, and a five figure harmonic of 17025, the latter of which is the fundamental basis of Cathie's unified equations.

In order to account for this, one must set aside The Bridge to Infinity, and turn to another of his books for the answer: The Harmonic Conquest of Space.

Now in this latter publication (his final book) there appears to be no reference at all to the stated mass harmonic of 170180681200, and no mention of a nautical mile of 6076.819455305 feet, direct or implied. Rather, in The Harmonic Conquest of Space, Cathie appears to have adopted a fixed nautical mile of exactly 6076 feet for his calculations. One can cite his use of this value in particular on page 36 [4].

By use of a nautical mile of 6076 feet precisely, Cathie is thus able to derive an earth volume in cubic nautical miles, that does indeed very closely match the numeric sequence of 17025. One can see how this is achieved below:

  • The radius values of the earth in nautical miles of 6076 feet are determined as follows (replicating Step 3 above):

                         Equatorial radius: 3963.190591704 / (6076 / 5280) = 3443.983924324 nm
                                 Polar radius: 3949.902764022 / (6076 / 5280) = 3432.436898294 nm

  • Based on these values, the cubic volume of the earth using Cathie's preferred 'ellipsoid formula' is also determined:

With this, one can be fairly satisfied then how Bruce Cathie derived a value for the volume of the earth, whose opening numeric sequence is closely tied to 17025. Essentially, when Cathie wrote The Bridge to Infinity, he made use of the value 170180681200, believing it to be correct based upon the nautical mile associated with the earth, as linked to this value.

However, later on when he wrote The Harmonic Conquest of Space, he appears to have completely discarded the value 170180681200, having 'lost confidence in its validity,' because he chose then to adopt a nautical mile of precisely 6076 feet for his calculations. And it was this, that modified the final answer for the earth volume, to make it much closer to the 17025 sequence.

The Speed of Light Harmonic

Concerning the speed of light, Cathie states that a value of 14389 is itself representative of a 5 figure 'speed-of-light harmonic.' Now a basic evaluation of the speed of light would indeed appear to offer a ready match to this value, but only when one chooses a very specific series of units to express the measure of speed.

The first of course, are Cathie’s preferred units of measure for distance: nautical miles i.e. units of 6076 feet. The second unit of consequence is the time unit. Now here Cathie makes use not of standard seconds, but rather a value of 8/9th of 1 second, which he calls 'grid seconds.' This indeed is a peculiar preference, yet one that Cathie does believe to be valid. To be sure, one will only derive the 'light harmonic sequence' of 14389 if one employs these units:

                                    Speed of light = 299792.458 Kilometres / sec.

                                    299792.458 × 0.6213711922 = 186282.397 Statute miles / sec

                                    186282.397 / (6076 / 5280) = 161878.054 Nautical miles / sec

                                    161878.054 × (8 / 9) = 143891.6035946 Nautical miles / grid sec
                                    Compare with 14389

Again, one may readily perceive that the calculated value for the speed of light as given in nautical miles per grid second, is a close match to the 5 figure value of Cathie.

The Unknown Harmonic

The value that Bruce Cathie initially refers to as an unknown harmonic, of 2636, was found by him to be derived from a relatively simple manipulation of the speed of light. The procedure for generating this value is contained in his basic mass equation, as follows:

                                       1 / 143891.6035946 = 0.0000069496758324

                                       Square-root of 0.0000069496758324 = 0.0026362237827030

                                       Compare with 2636

The Basic Mass Equation in Summary

Now in view of the above, the fundamental issue here, is quite simply whether or not Cathie's Mass equation is valid. To understand this central point, one must grasp just what exactly Cathie is actually saying with respect to his equation:

m = c + √(1 / c)

Basically, he is suggesting that purely by using the speed of light - as expressed in certain special units - one can derive a value for the volume of the earth, and that this value is so fundamental AND universal, that the relations expressed by it replace the M value in the primary Einstein energy equation, and that the proof of all this rests upon the addition of just two key values. They are as follows:

1) The speed of light, as expressed in distance units of nautical miles and time units of grid seconds, multiplied by 1000000:

                     143891.6035946 x 1000000 = 143891603594.6

2) The squareroot of the value: 100000000000000000000000000 divided by the speed of light:

                     √(100000000000000000000000000 / 143891.6035946) = 26362237827.0

By adding both terms together one has the following:

                     M  =  143891603594.6 + 26362237827.0 = 170253841421.6 cubic nautical miles

Now given this, one can see then that the final value of M does offer a numeric sequence whose first five numbers match 17025, which is the 'earth mass harmonic,' as specified by Cathie.

Objections to Cathie's Harmonic Unified Field Equations

In his books Bruce Cathie presents 3 equations that he calls Harmonic Unified Field Equations. They are all variations though on the basic Einstein energy equation, wherein he substitutes mass for: c + √(1 / c). If this substitution is in error, then all of the Unified Field Equations he presents must themselves be in error. Now in the evaluation that follows, both the Mass substitution component of his equations will be examined, in addition to one of his more elaborate equations: Harmonic Equation 3. This will help to isolate the main points of contention in Cathie's work. They begin as follows:

1) The Equations are Essentially Incomplete

It might interest the reader first just to consider attempting to solve Cathie's equation: mass = c + √(1 / c) to determine a value for Mass (M), by making use of the speed of light, as expressed in nautical miles per grid second. In doing so, by adhering to basic mathematical conventions, one obtains the following:

                                   1 / 143891.6035946 = 0.0000069496758324

                                   Square-root of 0.0000069496758324 = 0.0026362237827030

                                   143891.6035946 + 0.0026362237827030 = 143891.606230891

                                   Therefore: M = 143891.606230891

As one can see in this example, the end result is what would appear to be simply an alternate value for light speed that is only just slightly greater than the original value used at the outset. A value representative of the Earth volume as one would hope to achieve e.g. 170253841421.6, is simply not generated. And the reason for this is quite obvious.

The initial figures as Bruce Cathie presents them in his books only return the desired answer when the values of interest are heavily modified, such that the decimal points are in the ‘correct’ places before one solves the actual equation itself. Consequently, were one thus to use a speed of light value in Cathie’s basic mass equation, with the intention of producing an Earth volume containing Cathie’s noted 17025 sequence, the basic mass equation itself would have to be in the following form:

Only with this precise equation will one get two terms, which when added together, will equal the cubic volume of the earth in nautical miles, represented by M:

                                                 M  =  143891603594.6 + 26362237827.0

                                                 M  =  170253841421.6 cubic nautical miles

The above analysis draws attention then to an important point, which is that the basic mass equation as stated by Cathie in his books is not as simple as it appears. Extra manipulations 'internal' to his equation are necessary in order to solve it to get the 'correct' answer. Moreover, as all of his 3 noted Harmonic Equations have embedded into them the basic mass equation, this consideration applies to them also. One may cite indeed a very specific example of this 'complication,' as given in The Harmonic Conquest of Space. One that provides clear evidence of additional manipulations being required to solve Cathie's 'Harmonic Equation 3:'

Now this particular equation is essentially the basic Einstein energy equation with Bruce Cathie's mass substitution, with additionally, all of the C values being doubled (they are all multiplied by 2). Further to this, 'E' itself is ultimately obtained by taking the square root of the resolved components.

To emphasise then the use of built in extra manipulations to Cathie’s Harmonic equations, one may cite Cathie’s use of this very equation on page 26 of The Harmonic Conquest of Space. In the example given, in generating a value for ‘E’, Cathie employs a speed of light value of 143838.1966 nm / grid second. Multiplying this initial value by 2 to produce 287676.3932, he introduces the doubled light value into his equation on page 26 as follows:

From this initial line Cathie goes on through various workings to eventually end up with an answer of:

                                                E  =  26944444…

Now in reading through his particular workings here, were one to attempt to try to obtain this very answer by starting out from the initial line of the equation, as Cathie provides, one will certainly not arrive at 26944444… as a numeric sequence. And the reason, once again, is that in order to actually arrive at such a figure (as Cathie does) one must carefully and selectively make use of the correct '×10' multipliers to modify the various light speed values prior to either adding them together, or before taking their square-root.

Now this can only mean one thing: the equations that Bruce Cathie presents in his books are at best incomplete; being only partically correct. And at worst, they are completely erroneous.

2) Use of an 'Alternative' Ellipsoid Equation for Calculating the Volume of the Earth is Suspect

From the figures as presented it is quite clear that if the standard formula for an ellipsoid volume was used, then the accuracy of the match between the relevant harmonic values would be severely compromised.

Indeed, the ellipsoid formula of 4/3 × PI × a × a × b generates a value for the volume of the earth that is significantly different to that obtained by Bruce Cathie, using a modified formula based upon averaging the radius of the earth. There is in fact no real independent support for the use of the alternative ellipsoid formula. Purely because it helps to produce a volume value whose numeric sequence is close to 17025, it is simply assumed to be correct.

3) No Real Support for a Precise Nautical Mile of 6076 feet

It is very interesting when one reads Cathie's books to find that throughout all of his decades of research he never really appears to have answered the question - even to himself - just what is the correct and true value of the nautical mile?

At times he has used: 6076.819455305 feet. At other times a 'sea mile' of 6080 feet. Finally settling on 6076 feet precisely in The Harmonic Conquest of Space. This last value though, which indeed he appears to regard as the 'correct' one, appears to be so purely because when used in conjunction with the modified ellipsoid formula, he is able to derive a value that most closely matches the 17025 value, that helps to validate his Mass formula: M = c + √(1 / c).

There is no real independent confirmation as such that a value of 6076 feet for the nautical mile, is somehow correct.

4) The ‘M’ in Bruce Cathie’s Unified Field Equations is Unlikely to Represent Mass

Respecting Bruce Cathie’s proposed equation for mass – the fundamental basis of all of his noted equations – one is bound to wonder if the true identity of the variable ‘M’ as given, is indeed mass. Admittedly, mass as a physical concept is hard to pin down. However, though physicists themselves may disagree somewhat as to what exactly mass is, they pretty much do all tend to agree quite well on what mass is not. And in the realm of physics, mass is not volume; a fact which is evident in light of the recognised equation for density, which holds that density = mass / volume.

However that may be though, in Bruce Cathie’s basic mass equation, the actual value he produces for the cubic volume of the Earth is often referred to many times in his works as a ‘harmonic of mass’ (e.g. The Bridge to Infinity, p.40). But how can this be? In light of the relevant facts it does not appear that it can be. Rather, what he describes as mass would seem to be nothing more than simply volume. And indeed, that such a point is raised leads on to one very important further consideration: the issue of the equation actually balancing itself in terms of its physical units.

Under careful scrutiny it is easy to perceive that on one side of the mass equation there are two instances of speed (of light), whose physical units are both distance / time. But on the other side of the equation one has volume; an expression of distance, cubed. As a result, the overall equation just does not appear to balance in terms of its actual units: two instances of speed (albeit one slightly manipulated), equating to an instance of volume. That this would appear to be so is a strong indicator that the Unified Field Equations of Cathie either lack true physical validity, or that they are valid, but that the variables themselves are not exactly what they appear to be.

An Overall Assessment: Are Bruce Cathie's Harmonic Unified Equations Valid?

Given the above, it would seem that the evidence clearly supports the view that Bruce Cathie’s equations are in all likelihood completely wrong; being in essence a grand mathematical fiction. The main reasons are summarised as follows:

1) They require additional 'internal manipulations' to generate desired values of significance, and thus certainly cannot be complete as expressed by Cathie in his books.

2) The units of the basic mass equation do not balance, making it physically suspect.

3) The 'earth mass harmonic' that Cathie believes is tied to the speed of light, is generated both by an ellipsoid formula and nautical mile unit, whose selections cannot really be justified at all as such.

4) The suggestion that a numeric relation tied to the volume of the earth somehow generates a solution for mass that is universal, and thus can be a real substitute for mass in the basic Einstein energy equation, appears to be completely untenable.


[1] The Three books referenced by Bruce Cathie:

             The Bridge to Infinity: Harmonic 371244
             Adventures Unlimited Press (1997)

             The Harmonic Conquest of Space
             Nexus Magazine (1995)

             The Energy Grid: Harmonic 695, the Pulse of the Universe.
             Adventures Unlimited Press (1997)
[2] The Bridge to Infinity, (1997) p.23       
[3] Ibid, (1997) p.80   

[4] The Harmonic Conquest of Space, (1995) p.36