Due to the fact that illnesses spread most widely from person to person in the first fifteen days of August, it became a custom in Constantinople to bring out the portion of the Holy and True Cross of Christ kept in the Royal Palace and process it throughout the City of Constantinople from August 1 - 14. By experience the people of Constantinople knew, both government and ecclesiastical bodies, that the True Cross contained healing properties that was also able to rid all manner of illnesses and diseases merely through its presence.
We also have testimony that the Cross itself gave out a fragrant holy oil that was used to cure people, as well as the fact that the oil from vigil light that always burnt in front of the Holy Cross had miraculous properties by the grace of God.
The first comes to us from St. Bede, in chapter 20 of his Book of Holy Places:
Constantinople is bounded on all sides except the north by the sea, which extends from the great sea to the walls of the city, sixty thousand paces, and from the walls to the mouths of the Danube, forty thousand. The circuit of the walls, which are angular, according to the line of sea, is about twelve thousand paces. Constantine was at first disposed to build it in Cilicia, near the sea which separates Europe and Asia, but on a certain night all the iron tools were carried away, and, when men were sent to fetch them, they were found on the European side: for there it was the will of God that it should be built. In this city is a church of wonderful workmanship, called the Church of Hagia Sophia, built up from its foundation of a circular shape, domed in, and surrounded by three walls. It is supported to a great height on columns and arches, and has in its inmost part, on the north side, a large and beautiful closet, wherein is a wooden chest with a wooden lid, containing three pieces of our Lord’s Cross, that is to say, the long timber cut in two, and the transverse part of the same Holy Cross. These pieces are exhibited for the adoration of the people three times only in the year, namely, on the day of our Lord’s supper, the day of the preparation, and on the Holy Sabbath. On the first of these, the chest, which is two cubits long and one broad, is set out on a golden altar with the Holy Cross exposed to view: the Emperor first approaches, and after him all the different ranks of laymen, in order, kiss and venerate it: on the following day the Empress and all the married women and virgins do the same; but on the third day the bishops and different orders of the clergy do it, and then the chest is shut and carried back to the closet before mentioned. As long as it remains open on the altar, a wonderful odor spreads through the whole church. For an odoriferous liquor like oil flows from the knots of the holy wood, the least drop of which cures every complaint which a man may be afflicted with.
The second comes to us from St. Nikodemos the Hagiorite, in his Synaxarion under July 31:
Even the holy oil, which was lit in the oil lamp of the honorable wood of the Cross, worked miracles. Thus Christians would receive it, and they called it the Oil of the Cross. Wherefore Monk Cyril also writes that the venerable Savvas, with the oil of the oil lamp of the holy icons, worked many miracles and expelled many demons. In his history Theodoret writes that the venerable James, with the oil of the oil lamp, that was lit before the icon of the Martyrs (for which it was called the Oil of the Martyrs) healed many passions. Saint Thomais also healed a Monk who battled with fornication once she anointed him with the oil of her oil lamp.