Due to irregular spacing, character overlap, and varying orientations, text line detection is especially challenging for unconstrained handwritten and historical documents. These complexities make traditional methods, designed for straight lines, insufficient for detecting spiral text lines from Aramaic incantation bowls used in Sasanian Mesopotamia between the 4th and 7th centuries CE. We introduce a novel learning-based method for extracting spiral text lines inscribed on the surfaces of Aramaic incantation bowls. Our model utilizes an encoder-decoder architecture while leveraging connections among corresponding layers, similar to UNet. It combines high-level and low-level features, enabling precise localization and segmenting spiral text lines. Furthermore, we propose a novel metric to evaluate the dissimilarity between predicted and ground-truth lines. Inspired by the Intersection-over-Union (IoU) metric, We compare our model with the state-of-the-art methods and show that our approach outperforms these methods in terms of the introduced metric.