This chapter takes a step back and considers mathematics from a different angle: the mathematics closer to day-to-day practice. At the same time, the chapter brings forward the historical narrative by beginning to consider the science of the Roman era. The same duality will characterize the following (longer) chapter on astronomy, and both for a reason: in the centuries following the two breakout generations of Greek mathematics, most work in the exact sciences was concentrated on more “applied” mathematics and in particular on astronomy. It is therefore fitting that we begin with the author (not a mathematician but a historian) most emblematic of the turn from Greek to Roman: “Polybius, Introducing Applied Mathematics.”

This chapter is a kind of rondo, with Archimedes as its main theme. In outline, with five subchapters:

1. I. Prelude to Archimedes

2. II. Enter Archimedes

3. III. Response to Archimedes: Apollonius

4. IV. Archimedes’s Physics

5. V. Response to Archimedes: The Generation

A presentation of the defining characteristics of Platonic love, bringing in to focus those features which were most influential in subsequent ages. Of particular interest is the distinction between higher (‘ouranic’) and lower (‘pandemic’) love, as well as the notion of love as a mechanism for striving after the divine. The overarching narrative in which the development of Platonic love unfolds from the theological speculations of the Middle Ages to guides for the etiquette of conducting heteroerotic relationships during the Renaissance is presented.

Platonic love has often been attacked – notably by Vlastos – as fundamentally selfish. The basis of this argument rests on the claim that love for Plato is little more than a form of utility in which the beloved is exploited to enable the philosopher’s ascent and is simply the object of love to the extent to which he exhibits desirable qualities and not as an individual. This reading ignores the elements of erotic reciprocity found in both the Symposium and Phaedrus, in which the beloved is not merely a passive object, but rather an active participant with the lover in a mutually beneficial project defined by shared values, with both parties attempting to become as godlike as possible. The erastes is shown to care genuinely for the eromenos in the course of the philosophical pedagogy which Plato advocates. In the Phaedrus, the tripartition of the soul seems to be a further move towards non-individuality and the significance of the beloved’s individuality is regularly downplayed. Yet the abstraction inherent in Platonic love is revealed to be both beneficial as a mechanism for avoiding the negative aspects of interpersonal love, as well as a natural feature of the human condition: the striving after Beauty.

This chapter surveys different aspects of the theme of love (erōs) in Plotinus’ philosophy. Starting with what Plotinus finds significant in the human experience of love, we consider Plotinus’ nuanced evaluation of various types of earthly love (including sexuality), moving then to love as a desire of the beautiful expressing the very nature of soul in its relation to its origin in a divine transcendent Intellect, itself constituted in a relation of love to the ultimate first principle, the One/Good. Plotinus’ claim that the One is love/self-love is examined and two aspects of love, as expressing deficiency and as a generosity manifesting fulfilment, are discussed in relation to the One and as found in Intellect and in soul.

The influence which Plato’s theories of eros and philia exerted on Plutarch’s Advice to a Bride and Groom, On Isis and Osiris and his Dialogue on Love (Erotikos/Amatorius) are examined in light of the enormous chronological, cultural and philosophical distance between Plato and Plutarch. Plutarch’s methodological approach, reconciling his own opinions with the diverse views – which he saw as authoritative – expressed by Plato in the Symposium, Phaedrus and Laws, is considered. In his Advice to a Bride and Groom, Plutarch’s presentation of a philosophically informed love within marriage is presented as a counterpart to the early stages of Plato’s lover. Similarly, On Isis and Osiris, also presents a ‘companionate marriage’. However, the Erōtikos is the most important Plutarchan work dealing with Platonic love. Here Plutarch must prove not only that married love is natural, and homosexual love against nature (though he does not condemn it as a general principle), but also that women are worthy of love, and, if not already virtuous, can through the influence of the lover be brought towards virtue. The extent to which Plutarch rewrites the Platonic dialogues on eros in a new way, set in the imperial Roman world, is demonstrated.

Love and rhetoric are two major topics in Plato’s Phaedrus, unified by the dialogue’s theme of psychagōgia, or soul-leading. In Socrates’ great speech, the Palinode, Socrates explains how properly directed eros can assist souls in becoming the types of souls who will be capable of ascending closer to the forms and who will know themselves better. The Palinode is also itself an example of Socratic soul-leading with Phaedrus. Socrates acts akin a true lover to Phaedrus, in leading him towards philosophy and a philosophically informed view of rhetoric.

Dionysius endorses the Platonic position that love is a human response to beauty, and furthermore that the human quest for contemplation involves purification both of our understanding of beauty and of our experience of love. Dionysius’ understanding of both beauty and love goes beyond what we find in Plato; some of these changes belong to the development of the Platonic tradition, and especially to Plotinus, others to the Christian tradition. In the former case, beauty is understood less in terms of symmetry and more in terms of transparency to higher realities. This leads in Dionysius to a sense of beauty being a means of Theophany: God is manifest as beauty cascading through the different ontological levels of being. It is, however, crucial for Dionysius that this manifestation of beauty is seen in the way in which God’s love for all that exists is communicated through the ranks of being; this dimension he owes to his own Christian tradition. This enables Dionysius to integrate this understanding of being drawn to God through beauty with his understanding of the Church, both celestial and earthly, as essentially hierarchical, in which purification is experienced through participation in the rites of the Church.

Taking its start from Ovid’s Ars Amatoria, this essay explores various possible antecedents in the philosophical tradition, Platonist and Stoic. It argues for a process of formalization, in the Hellenistic era and later, of the criteria for ‘noble’, or philosophical, erotic relationships, to be derived from such dialogues as Lysis, Symposium, Phaedrus, and, not least, Alcibiades I, based on Socrates’ claim, at the end of the Phaedrus, that he possessed an erōtikē tekhnē. Such a tekhnē would have theōrēmata, and these were listed as (1) selecting a suitable love-object, or axierastos; (2) commending oneself to him; and (3) guiding him to moral and intellectual self-improvement. I suggest that Polemon, in the Old Academy, may have been instrumental in developing such a theory, but we find it formalized only much later, in the Didaskalikos of Alcinous, while further information is forthcoming from the Platonic commentaries of Hermeias (on the Phaedrus) and Proclus (on the Alcibiades). I argue that the Alcibiades seems to assume a more central role than the other dialogues in later Platonist theory, and acknowledge the important role of the Stoics in its formalization.