foreign materials, including elephant ivory and amber (41r–42r). The table of real materials that concludes the treatise confirms Nian’s utilitarian motivations as presented in the preface, transforming the abstract logarithmic calculus applied to polyhedron proportions into a practical skill inseparable from understanding real objects and substances in the reader’s world.
The ability to quickly calculate or even simply gauge the volume of a substance in a container, and therefore the value of that substance, would have been essential practical knowledge for both a Jingdezhen superintendent and a Grand Canal customs commissioner. After nearly a decade in each of these positions, Nian must have had considerable
experience in such estimates and calculations: perhaps the valuable substances he includes in the tables concluding the Brief Guide reflect that experience. He also would have needed to quickly calculate the material needs for porcelain production, such as amounts and costs of clay and pigments, the number of pieces that could be fired at one time in the kilns, how many pieces could be loaded on to a ship for export, and so on.35 With such a “practical studies” approach to understanding the mathematics of three dimensions, Nian’s approach to illusionistic Western painting is therefore precisely what one would expect from a mathematician. The Brief Guide and The Study of Vision are thus two approaches to the same subject: explaining and transmitting the technical knowledge required to understand and represent three-dimensional objects, although one representation is numerical and the other pictorial. To paint those objects in the same three-dimensional way that he perceived them, Nian simply needed to apply many of the same mathematical concepts to images. Nian’s two 1735 treatises, published only three months apart, demonstrate how he integrated his personal interests in Western mathematics and art by applying the precision and procedures of mathematics to painting by using measurements, calculations, and technical drawing tools. As distant as logarithmic calculus might seem from perspectival illusionistic painting, in Nian’s presentation they are two sides of the same coin, fundamentally related approaches to understanding one’s cognitive experience of the phenomenal world.
The Study of Vision
In the final lines of Nian’s entry in the Biographies of Astronomers and Mathematicians, Ruan Yuan writes that among Nian’s papers at his death was “an anonymous manuscript, entirely handwritten, with text as well as pictures [tuhua], which is also extremely elegant [bing ji jingmei],” which Ruan suspected that Nian had authored. Vague though Ruan’s comment may be, it notably does not describe the manuscript as having specifically mathematical content, and includes the tantalizing clue of illustrations alongside the text. It could not be the illustrated Brief Guide, which is listed individually. This mysterious manuscript, therefore, may well have been one of Nian’s two illustrated nonmathematical publications, which are not listed in the biography: The Essence of the Study of Vision and The Study of Vision. The latter has survived intact, but the former, an early shorter version of the same treatise, exists only in one fragmentary copy.36 That does not diminish the fact, however, that Nian twice published on the subject of pictorial illusions, demonstrating his enduring interest in the topic.
The Study of Vision comprises the original 1729 prose preface and a new additional preface dated to 1735, followed by 141 pages of illustrations that Nian refers to as tu. This term has been translated as “technical images” for the nonaesthetic knowledge encoded in such pictures, and tu have been described as “templates for action” that functioned together with nearby text to produce technical knowledge.37 Such images, which can be
found in woodblock-printed publications on all manner of subjects, inherently extend into the realm of art and art history because they are visual material with information to be interpreted beyond mere subject matter. Tu are often contrasted with more specific words such as “picture” or “painting” (hua) and “image” or “portrait” (xiang), which typically connoted works of art in contrast to diagrams, but none of these terms has a simple definition.38 In the late imperial period, hua and xiang were often used interchangeably to describe the illustrations in woodblock-printed literature, works with aesthetic effects on the viewer or reader. However, similar or even identical illustrations found in other printed contexts, such as anthologies of moralizing stories or biographies of exemplary figures, were often referred to as tu because of their functional value as didactic materials.39 Nian’s tu were intended to be used as templates for producing Western-style paintings, but do not themselves make any visual claim to be paintings. Instead, these illustrations and diagrams convey the technical basis of the relationship between representation and visual perception that was required to produce illusionistic paintings.
The eight sections of illustrations begin with the basic geometry required for drawing architectural features and spaces with linear perspective, and gradually introduce more complex forms and spaces before culminating in fully three-dimensional figures that cast shadows. Nearly all of the very limited prose text that inconsistently accompanies the illustrations is made up of straightforward step-by-step instructions for drawing lines between certain points using a straightedge in order to create the represented figures. The repetitive precision of these brief passages gives the entire treatise an extremely technical feel, in line with Nian’s other publications as well as the mathematical basis of linear perspective. By at least the seventeenth century in Europe, artists who were capable of using perspective scientifically, and whose work therefore seemed to connect the space represented in a painting with the space occupied by a viewer, were distinguished from those with an incomplete grasp of the “scientific rigor” required to achieve such effects.40 Given Nian’s personal devotion to mathematics, it is unsurprising to find the same rigor and exactitude in both the instructions and illustrations of The Study of Vision, although this is not a common feature of huapu.
Such technical precision, as well as the predominantly pictorial rather than textual nature of the treatise, is immediately visible in Nian’s first image (figure 2.1), a faithful but simplified outline mirror-image copy (perhaps traced?) of the first plate from Andrea Pozzo’s Perspectiva pictorum et architectorum (figure 2.2). The image depicts the artist’s tools required for perspective, or, from Nian’s view, those necessary to create European- style paintings: a drafting board with paper, compasses, straightedges, T-squares, and technical drawing pens in an ink pot. Immediately differentiating The Study of Vision from standard late imperial Chinese painting practices, in which the paper lay flat on the table, the upright position of the painting surface in this illustration also implies a difference in the pictorial outcome. The quality of the lines used to create this illustration is representative of those found throughout the treatise: all are fine, uniformly thin, and unmodulated.
2.2First plate from Pozzo, Perspectiva pictorum et archi-tectorum, English and Latin edition, 1707.
Straight lines are perfectly straight and right angles perfectly cornered, demonstrating that they were produced with the straightedges and T-squares seen in the illustration, while the open compasses indicate how perfect circles and precise angles are produced. Compasses and squares were not European imports, however; they are frequently seen in ancient images of Fuxi (holding a carpenter’s square) and his sister/wife Nüwa (holding a compass), the mythical beings who created the Chinese people.41 Straightedges (jiechi or zhichi) and line brushes (jiebi) were historically used to produce the precise straight lines in jiehua architectural painting.
Although an architectural painting specialist would have used tools such as straightedges in his work, literati painters did not (or did not admit to doing so). Paintings produced
