The quarter day and its correction
The Earth completes a circuit of the Sun in about three hundred and sixty five and a quarter days. A calendar year of three hundred and sixty five whole days is therefore slightly short, so on any fixed date the following year the Sun has not quite caught up: it sits about a quarter of a day, near a quarter of a degree, behind where it was.
Three of those in a row put the position about three quarters of a degree back. Then February gains a day, the calendar runs a day long, and the Sun on the same date is nearly three quarters of a degree ahead of the previous year. The lag is repaid all at once and the cycle restarts, which is why the figure circles a band rather than sliding through the sign.
What is left over
The correction is not perfect. A tropical year is a little under three hundred and sixty five and a quarter days, so the leap day slightly overcorrects, and the century rules in the Gregorian calendar exist to absorb that residue over much longer spans. Within a hundred years the leftover shows up as a small, slow slide of the whole four year pattern, which is part of why the range on this date is about one and a third degrees rather than exactly one.
The Sun also moves at an uneven rate through the year, faster near perihelion in early January and slower near aphelion in July. October falls on the accelerating side, which makes the autumn signs slightly shorter in days than the summer ones and gives their boundaries a rhythm of their own against the calendar.
What this means for the birthday
It means the date is a reliable coordinate. Anyone born on 10 October in any year of the last century has a Sun within a degree and a half of anyone else born on it, so the position, the decan and the aspects available to that degree are broadly common ground. That is a more substantial statement than the sign name, which only says the Sun was somewhere in a thirty degree stretch.
Seventeen degrees of Libra sits in the sign's second decan, the band from ten to twenty, given to Saturn in the Chaldean order beneath Venus as ruler of Libra. Because the whole century of variation stays inside that band, the decan is fixed for this date rather than year dependent as it is for birthdays sitting closer to ten or twenty degrees.
Where the date stops being informative
The Moon covers about thirteen degrees a day and changes sign roughly every two and a quarter days, so the calendar square cannot settle it. The rising sign is untouched by the date altogether, since a full rotation carries all twelve signs across the horizon and only the hour and the latitude choose one.
Those two are where two people sharing this birthday actually diverge, and they are also where most of a reading is done. The Sun is the placement that varies least between people, which makes it the least personal thing a chart contains rather than the headline it is usually treated as. Knowing the degree and the decan at least pushes the description down to a band of ten days rather than a whole month.
Questions people ask about this
Does my birthday drift through the zodiac over my lifetime?
No. The leap year rule keeps the calendar in step with the orbit, so the solar position on a fixed date circles a band of about a degree and a half rather than moving steadily in one direction.
What sign is 10 October?
Libra, in every year from 1930 to 2035. The Sun sits near seventeen degrees of the sign, which leaves a wide margin to the Scorpio boundary later in the month.
Why is the range about a degree and a half rather than exactly one?
Because the leap day slightly overcorrects the quarter day, leaving a small residue that the century rules absorb over longer spans. Within a hundred years that residue widens the band a little.
Does the precession of the equinoxes affect this?
Not for tropical positions, which are measured from the equinox point itself and therefore move with it. Precession matters when comparing tropical positions to the visible constellations, which is a separate question.
Which decan does this date fall in?
The second decan of Libra, from ten to twenty degrees, whose Chaldean sub ruler is Saturn. The variation across the years is small enough that the decan never changes for this date.
How much does the hour of birth change the degree?
The Sun covers roughly one degree a day, so a birth at midnight and one at noon differ by about half a degree. That is enough to matter for a tight aspect and not enough to move the sign.