In the previous posts to this series, I set out to articulate a perspective on the opening of Euclid’s *Elements *as arising by abstraction of forms that arise from sensible perceptions of things experienced in the real world. I aimed to make the case that certain ones of his Definitions and the forms his Postulates take are justifiable and understandable in light of them as arising in this manner. In this concluding, I consider one further definition and postulate and then finish with how this perspective of the *Elements* may be seen as an expression of beauty in relationship to in the quest for human understanding of geometry.

To “round out” my discussion of arriving at the subject-matter of mathematics using the opening of Euclid’s *Elements* as a case study, I want at another item of reality that would have been common-place in ancient Greece:

From the reception of this wagon wheel by the senses, by abstraction from its image, we may form a line from the outer part. But how should this particular line be defined? Bernard Lonergan speculated that Euclid could have defined it as a “perfectly round line” in same spirit as in defining a straight-line as a line lying evenly between its extremes. Instead, Euclid makes a more implicit definition which would arise by abstracting from the wheel its full shape as determined by its form. From this perspective, what determines the shape of the abstracted outer line is the equality of the lengths of the straight lines abstracted from the spokes emanating from the point abstracted from the hub. This is what Euclid gives us as definitions

A circle is a plane figure contained by a single line [which is called a circumference], (such that) all of the straight-lines radiating towards [the circumference] from one point amongst those lying inside the figure are equal to one another.

And the point is called the center of the circle.

And a diameter of the circle is any straight-line, being drawn through the center, and terminated in each direction by the circumference of the circle.

Furthermore, the formation of a wheel is achievable once the hub and the spokes are determined. By abstraction, this determination is understood to say how a circle depends solely on the center and the radius as given. This Euclid does give as a postulate (Euclid’s Postulate III):

And to draw a circle with any center and radius.

I would like to point that Euclid’s approach to defining the circle is one which extends without change to define the notion of a sphere in a space (and notions of hyper-sphere in any conception of higher dimensional space).

With this perspective of mathematics that I have drawn thus far, I have set out to portray the subject-matter as coming to be as part of the enterprise of human understanding. As such, I have endeavored to distinguish mathematics as arising from “seeing the form” as opposed to a cold and austere formalism. To clarify this little more, I would like to examine some of the features that play a role in the arrival of understanding of mathematics through form.

Returning to the fence examples, prior to the formulation of definitions and first principles that arise through abstractions from the sensible imagery, there is seeing radiating from the whole a unity of the pertinent features and the *proportios *that pertain, caused from the active awareness of the form of the fence. This holistic awareness is as one seeing a darkened room in all its details and relations after a light switch is thrown on. This illumination or insight from which understanding arises enables the formulation of the definitions and postulates by seeing the ways the parts are formed into a whole, enlightening the *proportios *to be formulated. Following Aquinas, we will refer to this prior awareness and seeing of form, which is what irradiates *proportio *rather than the *proportio* itself*,* as *claritas* (Latin for clarity, or brightness). It is this *claritas *that underlies the arrival of human understanding in every act of coming to know. It is where the distinction lies between identifying mathematics as a strictly formalized subject-matter, built as a system of definitions, axioms, and demonstrated propositions, and identifying mathematics as a subject-matter pertaining to *proportios *of parts ordered and united as a whole in forms abstracted from things sensibly perceived and understood in a supervening act of *claritas, *subsequently organized and articulated in definitions, first principles, and propositions. Moreover, mathematics in its formalization cannot be an object of knowledge without an act of *claritas to *see the forms and enable understanding to unfold from the underlying formal causes. It should be noted that those objects which are formed from abstraction are either from things having those forms (e.g. a fence) or from symbolic or iconic representations of them.

The two modes in understanding mathematics that I have outlined thus far, *proportio *and *claritas*, are, according to Aquinas, two of the formal components of **beauty**. In the context of mathematics, these formal components arise from the form of the sensible thing perceived and understood from images obtained by abstraction. Because abstraction is an act of the intellect that separates what is united in reality, what is become understood from the abstract forms may not fully conform to the form of the thing being abstracted. For example, while understanding of straight-lines may arise by abstraction of one of our fence images, the actual fence, in a fuller consideration, will be made up of straight-line rails, when seen in a series, will form a line that is curved as it conforms to the contour of the geography of the land. Thus, the mathematical object that would more closely conform, by abstraction, to form of the fence is a line, made up of a series of straight-lines, (i.e. being a piece-wise linear curve, in modern parlance) which curves in accord with those contours. The degree of perfection in which the form an abstract object conforms to the form of the thing being abstracted from is measured in its *integritas*. This is the third formal component of beauty, according to Aquinas. It serves to be the means by which the form of a thing and the form obtained by abstraction may be properly distinguished, compared, and understood.

Moreover, as a work of geometry, the *Elements* can be understood as seeking to reveal the form of geometry *per se*. As such, there is a measure of *integritas* in this work in two ways. First, with its opening that properly orders the basic material and first principles it is then leads the student of the subject-matter to understand the evermore richer and complex *proportios* that are internal to the structure as they become irradiated to the student in moments of *claritas*. The *Elements* are designed not only to present the various tableaus are the subject-matter, but is rendered so that those tableaus occur in a proper order to prepare the student for deeper topics still to come. Finally, the overall structure of the *Elements* is designed to move the student from the basics of geometry *per se*, to the complex *proportios* of plane geometry, to, in the end, giving the student who has understood all that has come before in *claritas* to begin to learn how plane geometry provides an understanding of spatial geometry, capping the work with a marvelous geometric classification and analysis of the Platonic solids. Thus, this work may be viewed as designed to give the student of geometry the means to start as a novice and grow in learning the ever richer and ever deeper *proportios* of plane and then spatial geometry, with the ultimate end being the prize of seeing in *claritas* the wonderful geometric structure of the Platonic solids, so central to Plato’s cosmology as accounted in his work the *Timaeus*, all reflected from the very form of geometry *per se*.

A second way that *integritas* may be measured in the work of the *Elements* is that, from the perspective of the student, it provides the way and the means to not only learn the depths and riches of geometry, but also to become a scholar of the subject-matter. As a student of geometry, the form of geometry becomes the good that that student desires and therefore seeks to understand and discover. The work of the *Elements*then may be seen as an object of beauty, in rendering the figures and their *proportios* proper to geometry, but also bring pleasure to the student to the extent that she or he is able to move in ever greater understanding through that work. Furthermore, the *Elements* is a work that provides the means for the student to move on to make further discoveries beyond the work, yet the work is one that can be returned to for further illumination or inspiration. This can certainly be said to be true of such great luminaries as Archimedes, Galileo, Descartes, Kepler, and Newton. In seeing the form of geometry is the end of one’s desire and the work of the *Elements* as the means to move in ever greater understanding of the *proportios*the form reveals, it is by the *Elements* that the student is educated in order to gain in knowledge by assimilation to that form. This leads by the effort of that student in becoming evermore assimilated to the form to be open to ever greater *claritas* beyond the work and move from student to scholar in making his or her own discoveries and rendering their own understanding for aiding others to greater knowledge. This is a further way that the *Elements* can be measured to have *integritas*. Moreover, we can also agree with Aquinas in how a personal journey for understanding can be characterized with respect to goodness and beauty:

Beauty and goodness in a thing are identical fundamentally; for they are based upon the same thing, namely, the form, and consequently goodness is praised as beauty. But they differ logically, for goodness properly relates to appetite (goodness being what all things desire); and, therefore, has an aspect of an end (an appetite being a kind of movement towards a thing). On the other hand, beauty relates to the cognitive faculty, for beautiful things are those which please when seen. Hence beauty consists in due proportion; for the senses delight in things duly proportioned, as in what is after their own kind — because even sense is a sort of reason, just as is every cognitive faculty. Now since knowledge is by assimilation, and similarity relates to form, beauty properly belongs to the nature of a formal cause. (Thomas Aquinas, *Summa Theologica *I.5.4 *ad* 1)

It is in framing of mathematics as the activity of the human understanding of sensible things through their intelligible forms by abstraction, whether from the things themselves or from their symbolic or iconic representations, that grounds the formalization of its subject-matter, not the reverse. In writing the *Elements*, Euclid could be credited with understanding well the discoveries his predecessors made and developed. He wrote a masterwork which opened with terms and principles that could be understood from the sensible things around us, expressed in a simple and self-evident way, and proceeds by reason to reveal in an orderly yet clear manner the ever-deeper properties and relations of and between the figures of plane geometry and beyond. Being the pioneer of this approach to organizing a mathematical subject, he can be credited with producing the *Elements* after having seen through acts of *claritas* of the form of plane geometry the figures that are fundamental, the proper ordering of their *proportios*, and the manner in which to represent, articulate, and demonstrate in order to render a work of plane geometry with a high measure of *integritas*. Thus, you may agree with me that Euclid’s *Elements* is a true work of beauty. The legacy of this work, as it has been passed down in history, to me testifies to its deep importance for inspiring mathematicians through subsequent ages in their development and rendering mathematics. I would certainly agree with Edna St. Vincent Millay who wrote

*Euclid alone has looked on Beauty bare.*

*Let all who prate of Beauty hold their peace,*

*And lay them prone upon the earth and cease*

*To ponder on themselves, the while they stare*

*At nothing, intricately drawn nowhere*

*In shapes of shifting lineage; let geese*

*Gabble and hiss, but heroes seek release*

*From dusty bondage into luminous air.*

*O blinding hour, O holy, terrible day,*

*When first the shaft into his vision shone*

*Of light anatomized! Euclid alone*

*Has looked on Beauty bare. Fortunate they*

*Who, though once only and then but far away,*

*Have heard her massive sandal set on stone.*