This section is from the book "Anatomy Of The Arteries Of The Human Body", by John Hatch Power. Also available from Amazon: Anatomy of the Arteries of the Human Body, with the Descriptive Anatomy of the Heart.
The Epigastric Artery arises from the external iliac, usually about three or four lines above Poupart's ligament: from this origin it takes a direction forwards, inwards, and slightly downwards, crossing anterior to the external iliac vein: it next turns upwards and inwards, so as to form a curvature, the convexity of which is directed downwards, looking towards Poupart's ligament, and, in some cases, even sinking into the femoral ring- the concavity looks upwards, and lodges a cul de sac of the peritoneum. We next trace the artery ascending obliquely inwards, between the fascia transversalis in front and the peritoneum posteriorly, in order to arrive at the inner margin of the internal abdominal ring. In this situation the vas deferens hooks round it, having first passed upwards and outwards, to the ring in front of the artery; and then downwards and inwards, to the pelvis behind it. From the inguinal ring, the epigastric artery continues to ascend obliquely inwards, till it gets between the posterior surface of the rectus muscle and its sheath; this latter structure presents at its termination inferiorly a lunated margin more or less distinct, and it is corresponding to this situation that we find the artery of each side entering the sheath. Finally the epigastric artery terminates by anastomosing with the internal mammary artery.
The branches of the epigastric are: the spermatic or cremasteric branch, which descends with the spermatic cord, to be lost on the coverings of the testicle: a branch which crosses behind the symphysis pubis, to anastomose with a similar branch from the opposite side; and an obturator branch, which descends behind the transverse ramus of the pubis, to anastomose with the obturator artery. It also gives several branches to the oblique muscles of the abdomen, some of which are of considerable size, and fatal hemorrhage has been known to arise from a wound of one of them, in tapping the abdomen. Lastly, the epigastric artery terminates in anastomosis with the internal mammary in the substance of the rectus muscle as well as within its sheath.
The Epigastric Vein arises from the external iliac close to Poupart's ligament, and ascends on the inside of the epigastric artery; it then bifurcates, and the artery lies between its divisions.