Members of the NPA praise Dr. Lucas scientific work, reason enough to buy the first volume of his four volume work (The Universal Force). To orientate myself in any new work I usually read the opening and closing paragraphs of chapters and interspersed random bits that have caught my attention while scanning through the book. The typesetting and general layout was a bit confusing and inconsistent.

Nevertheless, the content is the essence, not the appearance. Lucas, in Chapter 4, makes his first important assertion, quote: *Equation (4-44) is the derived version of the electrodynamic force for an elastic finite-size charged particle to order in Galilean transformation. The factors typically identified with Special Relativity Theory are found to originate from finite-size electrical feedback effects, nonlinear effects, and conservation of energy and momentum.* The last sentence is ambiguous: Is dependent on only one of the three cases under certain circumstance, or dependent on varying combination of the three cases?

Let’s derive Equation (4-44) on Page 92, the page can be viewed by clicking on the thumbnail to the left. We start with the Equation (4-43) now renamed to (1) and (2) and we assume these to be correct.

(1)

(2)

Lucas states: *Now the total electromagnetic force exerted by the moving charge distribution on a test charge is using equation (4-26) for the Lorentz force* and begins Equation (4-44) with

(3)

and the confusion and obfuscation begins. We forgive the typo, the equation uses and not as stated in the preceding text. However, we cannot overlook Lucas’ gross mathematical mistakes. He develops (3) to

(4)

The term marked should not be multiplied with the unit vecor and the remaining terms, marked should be grouped with a scalar equal to the magnitude of , that is a scalar is multiplied by a vector. The vector product does not exist in mathematics! Furthermore, how the term is obtained remains a mystery to me.

The correct development of (3) is

(5)

using (2) to express in terms of

(6)

using (1) to express in terms of

(7)

(8)

evaluating the dot products i.e and , and grouping the vector quantities. Finally, also making the substitution to obtain

(9)

from which Lucas’ Equation (4-44), that is Eq. (4) above, cannot be derived!

Four pages on, Lucas writes: *The generalized potential energy corresponding to equation (5-1)* [which is the same as his (4-44)] * for the electrodynamic force that is accurate to order in the Galillean transformation is*

(10)

At this point I closed the book since the work that followed was based on this flawed mathematics, there was no need for further review. Mathematics is the final arbitrator; the Eq. (10), which is (5-4) on page 96, is also mathematically incorrect. One can only square real and complex numbers, one cannot square, or take square roots, of vectors!

Lucas’ booked is filled with these and similar errors, therefor I cannot take this book seriously and consequently recommend others to dismiss it too.

Chapters 4 and 5 of my book contain the derivation and proof of an improved version of electrodynamics. Anton Vrba comments that he only reads the opening and closing paragraphs of chapters in the derivation and proof of an improved version of electrodynamics shows that he is not capable of understanding a proper derivation and proof in science. More than 1000 professional scientists have purchased this book and none of them have had any problem in following the mathematics, because every step in the derivations are provided. Some typos have been found in the book and corrected over time, but none of them have invalidated the results.

In the derivation that Anton gives he leaves out the vector identities that are involved showing his lack of understanding of the derivation.

… and as such the more than 1000 professional scientist must have been equally disappointed as I was.

Lucas’ claim the I show a lack of understanding is contradicted by my clear step by step derivation (5) through (9) using vector identities.

I also remind the readers of Frobenius Theorem which clearly states that associative division algebras over the real numbers is isomorphic to either the Reals, Complex Numbers or Quaternions, i.e. that is dimension 1, 2 and 4. As such the square roots or squares of three dimensional vectors, as Lucas formulates in his equations, just do not exist!

I will leave it to others to judge who has the better understanding of vector algebra and of the distributive laws of mathematics.