(ProQuest: ... denotes formulae omitted.)This is a story of an astronomical "instrument" or "figure" - both names appear in the accompanying texts - that lived from about 1430 until 1550. Exemplars can be found on paper, parchment and thin cardboard, always bound in codices; I do not know whether the device ever appeared in brass. Although the conceptual structure of the instrument remained largely stable over its lifetime, its mathematical and graphical features refused to solidify into a standard form. In its variant versions, the instrument exemplifies, I will suggest, a spirit of innovation and experimentation in practical astronomy in a century not especially known for such virtues. The instrument also exemplifies the attempt, not uncommon in this time, to replace astronomical tables and the tedium of interpolation with instruments of circles, lines, threads and rules. Yet despite the gestures of experimentation and efficiency exemplified by this instrument, it would not survive into the second half of the sixteenth century. After 1550, printed tables and ephemerides and new requirements of precision apparently rendered the instrument superfluous.Unlike such well-known astronomical instruments as the astrolabe, quadrant, Ptolemaic ruler and torquetum, our instrument would never achieve a fixed nomenclature. Most texts refer to it simply as an "instrument ... for finding the true conjunction and opposition of the Sun and Moon". Its peculiar shape, early on, evoked several names. A Latin text from the 1440s names the device a Rauchfudalium, a strangely composite German-Latin term perhaps referring to a censer (Rauchturabuluml) with chains forming a trapezoid shape similar to our instrument. Another text in this same manuscript reports that the instrument is commonly called a "salt dish [salzuas]". A German manuscript, originating in Nuremberg around 1460, describes the instrument as a "salt dish or measure [saltzuas oder kuffel]". Peculiarly shaped salt dishes, combining triangular and circular forms not unlike our instrument, produced by Nuremberg goldsmiths may have prompted this name. Early printed texts call the instrument an instrumentum astrologicum, emphasizing its clientele, or an equatoria, linking it to the long-established tradition of using rotating disks or vovelles to represent geometrical models of planetary theory.1 Although in this article I shall simply call the device a "syzygy instrument", accentuating function rather than shape, we must not forget the visual appeal apparently exerted by its unique physical design during its early years.The Context: Computing in Mathematical AstronomyMedieval and early modern astronomers computed. They calculated positions of the planets, dates and times of eclipses and new and full moons (syzygies), changing lengths of day and night throughout the year, the direction of Mecca, dates of Easter and other moveable feasts in the Christian calendar, divisions of the Zodiac into the twelve houses of the horoscope, and much more. Following procedures established by Ptolemy, including his practice of working to a precision of arc minutes, many of these astronomers devised elaborate tables - rows and columns of numbers - to enable such computations to be done with minimal recourse to sexagesimal multiplication or spherical trigonometry. Casting a horoscope might require literally hundreds of individual computational steps, and astronomers tried to keep the labour limited to sexagesimal addition and subtraction and interpolation of tabular values. Tablemaking was thus essential in this computational culture, and tables soon circulated widely, ranging from Ptolemy's Handy Tables and more than 100 Arabic zijes to the Alfonsine Tables by the 1270s. As J. D. North has aptly noted, the history of the dissemination and use of these latter tables "is almost coextensive with the history of European astronomy as a whole, at least until the sixteenth century". …