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The bounds are a system of dividing each zodiacal sign into 5 unequal regions. The regions are called bounds or terms. Each of the five bounds of a sign is ruled by one of the five planets beside the Sun and Moon (i.e. the non-Lights).
There was some variation found in the ancient literature, in which bounds were of different lengths or ruled by different planets. For instance, check out the different table of bounds at Altair astrology. However, the so-called Egyptian bounds are independently attested in many ancient sources. Additionally, the Egyptian bounds have been found in early horoscopes recovered by archaeologist. The Egyptian bounds are also the system that I use in my own practice.
The bound ruler of a planet or point, such as the Ascendant, has an influence over the nature of that planet or point. The bound is link between the two. You can think of signs as the houses of the planets, and the bounds as their rooms. Additionally, the bounds are key to certain predictive techniques in Hellenistic and Persian astrology. For instance, the bound ruler of the directed Ascendant was an important time lord (the distributor).
You can find a great set of charts of the various systems of bounds on the Altair Astrology site. When I’m online and want to look up the Egyptian bounds that’s where I go, as it’s a clear presentation.
Egyptian Bounds are Babylonian Bounds
Based on recent evidence, it has been proposed that the Egyptian bounds are actually of Babylonian origin (follow link for paper). The proposal is based on the discovery of two cuneiform tablets in which the bounds of the signs were recorded. These tablets may date back to the 4th or 5th century BCE. The paper provides some great technical and historic background on the bounds and their variation in ancient literature.