Hypercomplex Numbers and Early Vector Systems: A History A Thesis Presented in Partial Fulllment of the Requirements for the Degree Master of Mathematical Sciences in the Graduate School of The Ohio State University By Nathan Bushman, B.S. Graduate Program in Mathematical Sciences The Ohio State University 2020 Master’s Examination Committee: James Cogdell, Advisor Herb (Charles) Clemens © Copyright by Nathan Bushman 2020 Abstract If one were to study mathematics without ever studying its history, they may be left with a rather skewed perception of how the discipline has developed. Vector algebra is a particu- larly good example of this. Students may be introduced to vectors as early as pre-calculus, and will certainly have become closely acquainted with them by integral and multivariable calcu- lus. They are an essential means of representing and working with certain quantities – veloc- ity, force, etc. And so one may be led to believe that vectorial ideas must have been incorpo- rated into mathematics long, long ago. However, the reality is quite dierent; it was actually not until the end of the nineteenth century that a vector system (or vector algebra or calcu- lus) closely resembling our modern one was found, and not until the twentieth that it became widely used. The object of this thesis is to explore the interesting history behind this fact. We trace the widening of the idea of ‘quantity’ from its conception in classical geometry and algebra to one that admits a vector. We explore early mathematical systems that dealt with vectorial ideas, especially W.R. Hamilton’s quaternions. We explain how our modern vector system developed from this. The matters of how new ideas arise in mathematics and science, how such innova- tions are received, and how they evolve, are discussed both implicitly and explicitly. ii This thesis is dedicated to my parents, Brad Bushman and Tam Staord, to my siblings, Becca and Bran Bushman, and to my closest friends, Danny Vincenz, Julia Kerst, and Sana Mirza, all of whom continually oer me unconditional love and support. iii Acknowledgments First and foremost I would like to thank Jim Cogdell, who advised me throughout the course of this project. His guidance and kindness were invaluable in the completion of these pages. I would also thank Herb Clemens, who not only served on my committee but also encouraged me to pursue mathematics beyond the undergraduate level in the rst place. Lastly, my grati- tude goes out to the mathematics department here at OSU, and in particular to the following instructors who helped me mature mathematically or supported me during this program: Elliot Paquette, Joseph Vandehey, and Jenny Sheldon. iv Vita May 2014 . Olentangy High School August 2017 to December 2017 . Student Teaching Associate, The Ohio State University May 2018 . B.S. Mathematics, The Ohio State University August 2018 to present . Graduate Teaching Associate, The Ohio State University. Fields of Study Major Field: Mathematical Sciences v Table of Contents Page Abstract . ii Dedication . iii Acknowledgments . iv Vita .................................................v 1. New Numbers . .1 2. Hamilton’s Search for Triplets . 14 3. Hurwitz’s Result and the Law of Moduli . 29 4. The Development of the Quaternions . 42 5. Grassmann and His Theory of Extensions . 56 6. The Emergence of Vector Analysis . 76 7. Summary of Study . 95 Bibliography . 100 vi Chapter 1: New Numbers “Mathematicians are perfectly accepting of new ideas, provided they are a century old.” - Unknown Although both negative and so-called ‘imaginary’ numbers are an indispensable part of modern mathematics, they are, historically speaking, exceedingly controversial objects; de- bates about their use in the mathematical community1 raged on until well into the 19th cen- tury [25]. But why is this? To understand the answer to this question, we must rst discuss in some length the nature of mathematics as a eld of study. It has been said that the rst ‘true’ mathematicians were the ancient Greeks. This is almost entirely due to the emphasis they placed on the theoretical foundations of their work. Sev- eral pre-Greek civilizations showed an aptitude in solving mathematical problems comparable to that of the Greeks,2 but none of these peoples concerned themselves with the concept of mathematical proof (the importance of which is dicult to overstate) as the Greeks did. Consider, for contrast, the mathematics of the ancient Mesopotamians, more specically the 1 Most of the references to ‘mathematics’ (and ‘mathematicians,’ etc.) in this thesis will actually be references specically to Western mathematics (and etc.). The author wishes to make clear to the reader that this focus is due only to the nationalities of the key persons involved in this study, and that the general omission of ‘Western’ is present only for the sake of brevity; neither is intended in any way to write o the rich history of mathematics in other areas of world, such as China and India. 2 In fact, it seems that Greek mathematicians borrowed heavily from the Babylonians (discussed later in the paragraph of the note) in several areas, most notably astronomy – see [7]. 1 Babylonians. The Babylonians apparently knew how to solve quadratic equations and related problems [23]. Additionally, plentiful records exist of extensive tables of squares of numbers, products of large numbers, and ‘Pythagorean triples’ developed by Babylonian scribes [13]. Yet for all these impressive results, no piece of Babylonian mathematics places direct emphasis on either the generalization or justication of results. As one mathematical historian wrote: “the Mesopotamians . addressed only the question of ‘how’ while avoiding the much more signicant issue of ‘why’ ” ([13], pg. 5). Some have argued that the vast number of solutions Babylonian ‘algebraists’ accumulated to similar types of problems points toward them having some rationale for their procedures, but this can only be treated as an implication of the his- torical record, and it remains that Babylonian mathematics was of a very prescriptive nature and hardly emphasized the notion of providing logical support for one’s methods [23]. Now it is well-known that early Greek mathematicians accomplished much beyond intro- ducing the idea of proof. The works of Euclid, Archimedes, and others yielded a bounty of in- tellectual fruit that is still admired today, as has been described by many authors across many writings in romantic detail. However, the object of the current study is not to add to this ample literature; the relevance of Greek mathematics to the story of negative and imaginary numbers is, perhaps unfortunately, relegated to the contribution of the tradition to their very belated acceptance. It is apparent that these prominent Greek authors thought of their mathematics as em- pirical. Though it is true that they dealt with abstract ideas of geometry such as “breadth- less lengths” and objects “with no part” [14], these notions were only used to create an ide- alized reection of their reality, and ultimately their axioms were derived from what they could clearly see with their own eyes. Hence, to the Greeks, mathematics was not, as we may think of 2 it today, a eld of study oering (among other things) support for the natural sciences: rather, it was one of these sciences. We can be condent in this both because a great many ancient Greek writings paint mathematics as a means of unearthing truths about the physical world [13] and also because Greek authors did not generally speak of purely abstract mathemati- cal ideas. To the latter point, we note that Greek authors did not speak of numbers in isola- tion, but rather quantities – lengths, areas, and volumes – the implication being that numbers on their own, not representing something anything concrete, merited no consideration.3 In this same vein, equations in the works of these authors were always homogeneous, that is all terms within were of the same degree – representing the view that one cannot add quantities of dierent ‘types,’ such as say one length and another volume, together [23]. This empirical perspective on mathematics was strongly inherited by the wider European traditions that succeeded the Greeks’. In fact, the core beliefs entailed by it came to be held much more consciously and rmly by the mathematicians of said successive traditions, likely due to the admiration many of them held for Greek mathematics; by the beginning of the early modern period, mathematics had come to be predominantly viewed as the science of quantity. And this brings us back to the subject at hand, for it is this point of view, more specically the exclusion of purely abstract mathematical ideas that it necessitates, that kept ‘new numbers’ from being accepted by mathematicians for so long [25]. Why this is the case quickly becomes clear when one adopts, for the sake of argument, the philosophy in question. Indeed, consider negative numbers from this point of view (note, this 3 Strictly speaking, this is only true starting around the time of Euclid. Pythagoras, who lived around two- and-a-half centuries earlier, had a philosophy heavily dependent on the abstract idea of numbers and also of harmony. He is said to have professed that “all things are number” [5]. Views such as this, though, were even- tually supplanted by the empirical idea of geometry that the ancient Greeks had [5, 13], and it is the ultimate mathematical philosophy of the Greeks (which would be inherited by further Western traditions) that we are con- cerned with here. 3 ought to be sucient, for if negative numbers are regarded almost as anathema to mathe- matics, what can be said of their supposed square roots?). We understand that negative num- bers have concrete meaning only in specic contexts: there are many problems in which they are nonsensical; “..