The catalogue called the 'Curiosum urbis Romæ' places not only the Baths of Titus but also those of Trajan in the third region. The anonymous MS. of Einsiedlen places Trajani Thermæ near the Church of S. Pietro in Vincoli; and Anastasius in his 'Life of Symmachus,' mentions them as near the Church of S. Martino. It is, therefore, abundantly proved that the Thermæ of Trajan stood at the back of the Baths of Titus, and it is here that we find them placed in the plan of Palladio. That they were distinct buildings seems clear from an inscription in which they are separately mentioned. A satisfactory explanation of the apparently strange fact that Trajan erected new and smaller Thermæ near those of Titus is given by one of the chronologers of the period, who speaks of the Baths of Trajan as intended for women, for whom there was no separate accommodation provided in those of Titus. The scattered ruins to the north of the Baths of Titus may have belonged to Trajan's Baths.
On the Esquiline Hill, besides the Baths of Titus, the Domus Aurea, and the Sette Sale, we find four remarkable ruins, which are called the Trophies of Marius, the Arch of Gallienus, the Minerva Medica, and the Auditorium of Mæcenas.
The ruin called the Trophies of Marius stands at the corner of the Via di S. Bibiana. It consists in the lower part of a number of small and curiously shaped compartments of brickwork, with openings at seven or eight different points. Underneath these, and now hidden under the level of the ground, is a large basin or tank, and above them the upper part of the building is formed by the remains of three niches, in which stood the marble trophies now placed upon the balustrade of the steps of the Capitol. They were removed to the Capitol by Sixtus V. in the year 1585. The name Trophies of Marius is an attempt to explain the more ancient name of Cimbrum, which we find attached to the ruin in the middle ages, by identifying the trophies with the Tropæa Marii mentioned by Suetonius as having been pulled down by Sulla and restored by Julius Cæsar.[74] But although we must allow that there is some probability in the supposition that the Marian trophies may have occupied these niches, yet it is certain that the building itself was intended to serve another purpose, that of the castellum or principal reservoir of an aqueduct, with a public fountain in the form of a cascade in front. The basin which has been discovered under the building, and the peculiar shape of the complicated interior structure, can be best explained thus, and the remains of some part of the aqueduct itself may be seen at the back. It was at one time supposed that the Aqua Julia ended here but it is now generally acknowledged that the ruin belonged to the Aqua Alexandrina, and that the name Nymphæum Alexandri, found in the catalogues of the fifth region, must be assigned to it. The Alexandrine Aqueduct was built by Alexander Severus in the year A.D. 225. Water was brought to Rome by means of it from a spot near the Lake Regillus, and a portion is still visible on the left hand of the Via Labicana about two miles from Rome. The level of this aqueduct corresponds exactly with the building in question, and the style of brickwork and architecture are such as might belong to the third century. It is possible, as Reber remarks, that Alexander Severus may have found the exact spot where the Trophies of Marius had been placed by Julius Cæsar convenient for the castellum of his aqueduct, and have used the trophies to ornament the new building which he erected.