A Philosopher Giving a Lecture on the Orrery is painted with various intended references -- it does resemble a conversation piece portrait, and many of the figures are based on real life groups and people (specifically, the Lunar Society and Sir Isaac Newton). It is also one in a series of "candlelight paintings" that Wright created that made use of tenebrism and emphasized scientific topics. This work, however, was painted in the 18th century.  The Renaissance period took place between the 14th and 17th centuries in Europe.

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Neoclassicism was an artistic movement developed in the late eighteenth century that greatly valued classical ideals in art, particularly in crafting highly structured compositions which fit very specific guidelines. Among these guidelines were a de-emphasized background, use of a central vanishing point that focused the painting on one spot, frequent use of overlapping figures in profile, stoic expressions on main figures, and a heavy reliance on straight lines. Among the key stylistic elements of Neoclassical was a desire to hide any and all brushstrokes, so as to make sure the viewer was not distracted from understanding the painting's message by the artist's physical work.

Artwork from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Jacques-Louis_David,_Le_Serment_des_Horaces.jpg

Both the philosophical movement called the Enlightenment and the artistic movement Neoclassicism, of which The Oath of the Horatii is representative, were both developed in France during the eighteenth century. Beyond geographical proximity, each movement also looked to Ancient Rome and Greece for inspiration, highly prized rationality over emotionalism, and favored strict order and logic. 

Artwork from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Jacques-Louis_David,_Le_Serment_des_Horaces.jpg

French for "deceive the eye," trompe l'oeil is the painting technique where the artist makes an image that appears three dimensional and outside the painting. The concept goes back to Roman art, where walls in houses would have fake windows of iterms painted on, but was first defined in the Baroque era. The technique can be used to either enhance the dimensionality of a painting, or as a kind of artistic joke, featuring a "mistaken" element.

Each painting features a remarkable realism in its portrayal of the figures included therein, including a self portrait of the painter. Francisco de Goya likely included himself in his painting as a nod to Velázquez' own self-portrait in Las Meninas, with both standing behind an easel placed in the left side of the painting. Both painters include themselves to make the viewer rethink what they see in the painting of royalty, with the gaze of the painter staring back at the viewer.

Figure 1: Las Meninas by Diego Velázquez (1656)

Figure 2: Charles IV of Spain and His Family Francisco de Goya (1801)

Chiaroscuro is an artistic term denoting a strong contrast between dark and light elements. First developed and defined in Italy in the late sixteenth century, chiaroscuro was a key element of Rembrandt van Rijn's style. In The Night Watch, he throws significant light on the group's leaders exiting the hall, with shadow taking over those in the back.

Figure 3: The Night Watch by Rembrandt van Rijn (1642)

Figure 4: The Astronomer by Johannes Vermeer (1668)

The "Cavalier" dress refers to men with flowing ringlets, high leather boots, brightly colored fabrics, lace collars, and elaborate trimmings. The word "Cavalier" described the Royalist factions in the English Civil War (1642-1651), although Van Dyck painted portraits of Royalist and Parliamentarian leaders before the war. In fashionable portraiture over the next centuries, especially in England, "Cavalier" style was a popular way to be presented.

The most famous "triple portrait" is the one that Sir Anthony van Dyck painted of King Charles I of England in 1635 or 1636. The painting was supposed to be used as a model for a bust, and the three different angles of Charles' face were intended to allow for a possible three-dimensional model. The painting became widely copied in the next few centuries, as it not only showed off the subject but also exhibited the skill of the artist.

An orrery was an early form of planetarium, and it was meant to show how the planets move in the solar system.  The light projecting from the left side was meant to represent the sun.  The philosopher is demonstrating what happens during an eclipse in this image.

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Wright uses tenebrism in A Philosopher Giving a Lecture on the Orrery. Tenebrism is using very contrasting lights and darks, to the point that the the darkness dominates the image. This then emphasizes the light spots even further -- in this painting, the parts lit by candlelight are very noticeable in comparison to the rest of the image. Wright painted several "candlelight pictures," and he was influenced by Caravaggio, who also made use of tenebrism in many of his works.

Image is in the public domain, accessed through Wikipedia Media Commons: https://en.wikipedia.org/wiki/File:Wright_of_Derby,_The_Orrery.jpg