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.

This chapter is introductory. I first survey, in a quick sweep, mathematics before Greece. This is followed by the historical context for the rise of mathematics in Greece itself (a discussion heavy with historiographical problems because so much is speculative). Finally, I conclude with a picture of the earliest known Greek mathematics.

Platonic and Plotinian conceptions of beauty played a role in Ficino’s understanding of beauty in De amore and subsequently in Leone Ebreo’s (Judah Abravanel’s) understanding of beauty in Dialoghi d’amore. Plato’s Symposium is an inspiration to both but does not determine the structure of either author. This essay contrasts Ficino’s presentation of seven friends’ viewpoints on love and Leone’s dialogue between Philo and Sophia, who together create PhiloSophy. While Ficino’s letters to Cavalcanti, to whom he dedicates the work, provide further documentation of Ficino’s intent to experience a higher form of love, Leone’s view of Aristophanes’ androgyne as based on a pre-Talmudic Hebraic concept of Adam as a hermaphrodite indicates his subordination of the Symposium to the Hebraic tradition.

The essay shows the influence of Plotinus’ idea of love of intellect alone for the heavenly Venus and the influence of the Middle Stoics and Middle Academics on Ficino’s belief that there are generative seeds of virtue and knowledge in the human soul. Alberti’s concept of concinnitas and measure are at the root of Ficino’s notions of harmony of sounds and harmony of sights. Leone, like Ficino, values the senses of hearing and sight and disparages the senses of taste, smell and touch.

Both authors correlate beauty and virtue. Ficino is a major source for the Renaissance tendency to view the picture of a beautiful woman as a picture of a virtuous woman. Sophia and Philo both grasp the relationship between beauty and virtue and that the highest goal is a love relationship with the Divine Creator; nevertheless, Philo resorts to lecherous longing for Sophia despite recognizing that coppulazione is the moment of a human’s intellectual vision of God.

Marsilio Ficino adopted the concept of love from Plato and the Neoplatonists in order to elaborate his philosophy of humanity and divinity. In commenting on Plato’s Symposium, he employed amor/eros for his philosophical theology according to which God and the universe are united: love is that which binds all levels together. As such a mediator and power, it engages humans as a species, as individuals, and in societies. Love is the formula that makes the metaphysical reality intelligible and teachable. Love divides the good from the abject and sets the standard that is attainable and binding for all.

A one-volume history of Greek mathematics has to be selective and compressed. All the more so, a one-chapter history of Greek astronomy. We follow an entire trajectory—from the beginnings to Ptolemy (second century ce). Chapter 3, “The Generation of Archimedes,” was a rondo with Archimedes as its recurrent theme. Here, we progress through three subchapters for our three main cast members:

1. I. Rising: Greek Astronomy to Eudoxus

2. II. Conjunction: Greece, Babylon, and Hipparchus

3. III. Culmination: Ptolemy

In a world of ambitious mathematicians, each vying to become the new Archimedes, Evangelista Torricelli, born in Faenza, Italy, in 1608, was determined to out-Archimedes them all. The one book he published in his short life (he died at age thirty-nine) is the Geometrical Works, a series of mathematical marvels, some reproducing results from Archimedes, some extending the spirit of Archimedes by measuring new curved figures. Torricelli may have been the first to measure the area of the cycloid, a figure originally proposed by Galileo. (The seventeenth century was full of Archimedes-like challenges; unlike in the third century bce, such challenges tended to generate many responses, with many ensuing fights over priority.) He invented and measured his own—paradoxical—figure. Consider a hyperbola, one of its asymptotes, and some perpendicular to that asymptote cutting the hyperbola. The three taken together define an infinitely long figure that is widest at its base (the perpendicular) and keeps narrowing—without ever reaching the zero width of a point—as it moves away from that base. Rotate this figure around the asymptote, and you get an infinitely long solid, shaped rather like an infinitely long, curved funnel. Torricelli measured the volume of this infinite solid—which turned out to be, incredibly, finite. Infinite surface, finite volume! A new Archimedes, right there!

This essay offers an account of Thomas Aquinas’s very complex understanding of love (amor) and related concepts, such as delight (delectatio) and divinely infused charity. It acknowledges the many Platonic influences on Aquinas’s thought in these regards, influences that come to him especially through Augustine of Hippo and Pseudo-Dionysius – both of whom he occasionally criticizes, however, for adhering too closely to Platonic doctrines. Aquinas is intent upon tying all human love in some sense to nature, acknowledging that this entails that in many regards human love is necessitated. To the extent, however, that the rational appetite (the will) is involved, love is not necessitated. The divinely infused virtue of charity brings in further complications, for it is a love that is not natural but comes to a person from above: it is supernatural. And yet it too, according to Aquinas, is not wholly necessitating but affects man in a way that leaves his will free.

This chapter explores the different treatments the topic of love received in the Symposium and the Phaedrus. The Symposium expressly addresses the question of the character and benefits of eros – and the question of its nature. It is a philosophical enquiry into this phenomenon, conducted by a set of mature men (and the memory of one exceptional woman) who have personal experience in this topic and have thought about it. In the Phaedrus, by contrast, the topic is addressed from the point of view of a would-be lover trying, through a speech addressed to the young and inexperienced man, to persuade him to submit to his overtures and desires. For Plato, eros is crucial to the practice of philosophy, a force which can take us in two separate directions, towards the good and towards the bad.