Restarting the Navitor: You'll get it by hook or by crook in Spring.
The puzzle: Get thee behind me and figure out the difference between then and now. Then light my fire!
1. Restart the Navitor.
It is now Spring (Ver), so on Kal. Mar. the Navitor breaks down and you're back in Cornelius's study. Go to the double doors at the back of the study (to the right of the Navitor) and click on the handle. A note appears under the door with a code number (59). Go to the Ver table (to the right of the Navitor), enter LIX on the box, and take the machine piece out. Insert the piece between the crank and the vents and run the machine by clicking on the handle and holding the mouse button down to make it turn. A slot will open along the table edge; pull the knob. Find the door marked "Ver" by going to the left of the Navitor, and then to the right. Go in, climb the ladder, and attach the hook to the bellows. The Navitor will now run; go back to it, pull the crank to the right, and find yourself back at the Arch of Severus.
2. The Vestal Flame (Id. Mar.)
The scrolls and Acta report the Eternal Flame going out. Across from the back of the Temple of Vesta is a set of doors that will be automatically open on or after Id. Mar. Go through and get the lens holder. Turn around and pull the lever by the door; a back door to the Temple opens. Slide the upper wooden knob to enter "Kal. Mar." -- the date of the old New Year and when the Vestals relight the flame -- at the top. With the lower knob, enter the number of days between Kal. Mar. and today (if today is Id. Mar., enter XV). Pull the lever; a little door at the bottom right opens to reveal a lens. Click on it and enter the front of the Temple. Attach the lens to the holder above the brazier; the flame lights up the Temple, revealing on the side a skull, a vial, and a scroll. Get them.