We describe a system for the automatic quantification of structure in spiral galaxies. This enables translation of sky survey images into data needed to help address fundamental astrophysical questions such as the origin of spiral structure---a phenomenon that has eluded theoretical description despite 150 years of study (Sellwood 2010). The difficulty of automated measurement is underscored by the fact that, to date, only manual efforts (such as the citizen science project Galaxy Zoo) have been able to extract information about large samples of spiral galaxies. An automated approach will be needed to eliminate measurement subjectivity and handle the otherwise-overwhelming image quantities (up to billions of images) from near-future surveys. Our approach automatically describes spiral galaxy structure as a set of arcs, precisely describing spiral arm segment arrangement while retaining the flexibility needed to accommodate the observed wide variety of spiral galaxy structure. The largest existing quantitative measurements were manually-guided and encompassed fewer than 100 galaxies, while we have already applied our method to more than 29,000 galaxies. Our output matches previous information, both quantitatively over small existing samples, and qualitatively against human classifications from Galaxy Zoo.