With the distinctions between absolute and relative, true and apparent, and mathematical and common understood as I suggest, I will now argue that they are empirically accessible, and therefore subject to empirical investigation. Moreover, this is something of which Newton was well aware at the time he was writing the Principia.
Our empirical access to time is indirect, via material bodies: we make use of material systems that exhibit periodic phenomena, that is, which tick. The most obvious periodic phenomenon in our daily lives is the cycle of day and night itself. Traditionally, and into the fifteenth century, the hour was defined by taking the time from sunrise to sunset and dividing it into twelve (and similarly for the twelve hours of the night). Call this seasonal time. Time understood in this way is relative (it depends on the relative motion of the Sun and the Earth) and apparent (deriving from the motion of the Sun as it appears to us).
How would the division of day and night each into twelve hours be achieved? During the day, the Sun appears to us to move steadily across the sky, so the apparent motion of the Sun (assumed to be constant) could be used to divide the day into twelve. During the night, the stars similarly appear to move steadily across the sky, so the apparent motion of the stars (assumed to be constant) could be used to divide the night into twelve. One upshot of this approach seems to be that daytime hours in summer are longer than daytime hours in winter. But how can that be ? We defined the hour as our unit of time, and by definition each hour must be of the same duration: an hour. So what does it mean to say that hours vary in duration, and how could we show that this is the case ? One way to tell is to compare these seasonal hours with other periodic phenomena, such as the number of times sand runs through an old-fashioned egg-timer (or an hourglass!) in the course of an “hour”; even our own biological rhythms are sufficient for us to be able to “experience” a difference in length of seasonal hours.
In other words, there are different apparent times, arising from different choices of bodies whose relative motions we use to construct a clock. Faced with such disagreement, it can be convenient for a society to make a choice that is shared among its members. We adopt a common time, and in the mid-seventeenth century, apparent solar time was used for this purpose. Apparent solar time is defined as follows: take one complete circuit of the Sun around the Earth and divide it equally into 24 hours, so that one hour corresponds to the Sun moving through an angle of 15°.
Apparent solar time is a relative conception of time (depending on the relative motion of the Sun and the Earth). During the day, it is directly observable (look and see how long it takes for the Sun to move through 15°, or measure this indirectly by means of a sundial). During the night, we need theory to calculate how far the stars have to move for a corresponding 15° motion of the Sun (postulated to be continuing its motion around the other side of the Earth). With this calculation in hand, such “apparent solar time” is observable—it is in the appearances and is therefore apparent.
When compared to other periodic phenomena, this is a great improvement: those phenomena are now in much greater agreement with our chosen clock. So there is good reason for ordinary people to switch from seasonal time to apparent solar time as the basis for common time, and many cities and towns did so. According to Audoin and Guinot (2001, 40), apparent solar time was in use as the standard for common time in country areas of Europe until the beginning of the twentieth century.
Astronomers too would have had good reason to prefer apparent solar time over seasonal time as their time parameter for astronomy. They were engaged in the task of predicting the motions of the heavenly bodies (the stars, the Sun, the Moon, and the planets) using uniform circular motion. Any deviations of these bodies from uniform circular motion needed to be accounted for theoretically. By adopting apparent solar time rather than seasonal time, the motions of the heavenly bodies become less irregular. That is to say, they are less irregular with respect to equal intervals of time defined using apparent solar time. The role of theory in our understanding of time is clear: if one choice of clock (the standard of time that we adopt) yields the result that the motions are highly irregular, while another choice of clock yields the result that the motions are close to regular, then the second clock makes our theoretical task easier as we try to account for remaining irregularities.
Despite its great practical utility as a basis for common time, apparent solar time was nevertheless deemed “irregular” even by ancient astronomers. They worked instead with mean solar time, which is constructed as follows. Every day, when the Sun rises, it rises at a slightly different point relative to the background of the stars. A solar year is the time taken for the Sun to rise again at that same point plotted against the background of the stars. In the course of this year, if we plot the position of the Sun at sunrise every day with respect to the stars, we see that the Sun makes a complete journey around the sky, along a path called the ecliptic. This path is so important that we keep track of it at night (when we cannot see the Sun) by means of the constellations of the zodiac. But the Sun speeds up and slows down during the year (passing through a greater or lesser angular distance along the ecliptic on different days). If we smooth out the motion of the Sun, so that it moves at a constant daily speed around the ecliptic, the resulting position with respect to the fixed stars is the position of the mean Sun. Except when the actual Sun and the mean Sun coincide, no material body is located at the position of the mean Sun. But, if we use the mean Sun as our clock, the motions of the heavenly bodies overall become more regular: regularities in the irregularities are removed. This serves our goal of constructing a predictively adequate theory using regular motions: irregularities in the appearances (the apparent motions) are to be accounted for by such things as the eccentricity of the Sun’s orbit, and so forth. Mean solar time “corrects” apparent solar time by removing periodic irregularities via the “equation of common time.” Notice that we are using theory here to move from apparent time, derived from the most regular apparent motions that we observe (celestial motions), to an abstract theoretical time. Mean solar time is a theoretical construct: no material bodies used in its construction were observed to move regularly with respect to mean solar time, and it was therefore neither apparent nor relative. Astronomers knew of no relative motions that could serve as a clock with respect to this time. Thus, the “time” that is appropriate for astronomy is mathematical, and it is neither relative nor apparent.
This is a conclusion that we arrive at through an interplay between theoretical and empirical considerations, in which a theory of planetary motions is constructed by assigning simple basic motions to those planets and then treating the deviations as corrections to those basic motions. The concepts of relative, absolute, apparent, true, common, and mathematical time were all present in the development of the project of constructing a time parameter appropriate for the purposes of mathematical astronomy, as that project existed at the time that Newton was writing, and therefore all three distinctions were empirically engaged at that time. These are distinctions that we can get at empirically, through the project of mathematical astronomy.
The intricacies concerning the treatment of time in mathematical astronomy were familiar to Newton. In the scholium to the definitions he writes,
In astronomy, absolute time is distinguished from relative time by the equation of common time. (Newton 1999, 410)
In other words, we move from common time (a time adequate for the purposes of regulating our daily lives) to a time parameter appropriate to the needs of mathematical astronomy, and in so doing we move from relative time, on which common time is based, to absolute time, making use of the equation of common time and therefore of the mathematical properties of our time parameter. Newton further elaborates on the role of the equation of common time as follows:
[D]uration is rightly distinguished from its sensible measures and is gathered from them by means of an astronomical equation. Moreover, the need for using this equation in determining when phenomena occur is proved by experience with a pendulum clock and also by eclipses of the satellites of Jupiter. (Newton 1999, 410)
The point he makes in this paragraph is that, in practice, we have strong theoretical reasons for believing that we have not yet found bodies (either celestial or terrestrial) whose periodic motions can serve as perfect clocks. The time parameter that would be measured by such perfect clocks, and that we have strong empirical reasons for adopting based on our investigations in astronomy and in terrestrial clock making, is neither relative nor apparent. Thus, Newton knew very well that the “time” that is appropriate for astronomy, and thus the “time” that is appropriate for solving the problem of the system of the world, is mathematical, and is neither relative nor apparent. In specifying his three distinctions, he knew that they are distinctions that we can get our hands on empirically, and that we do this via the theories that we construct in order to account for the detailed motions of celestial bodies (the planets) and of terrestrial bodies (pendulum clocks).11
-  For a detailed discussion of the historical and conceptual development of time measurement, see Audoin andGuinot 2001.
-  Notice that this attributes all the irregularities to the Sun. The motion of the stars with respect to the Earthis assumed to be uniform, and this is used as the regular background with respect to which the apparentmotion of the Sun is then smoothed out to construct the mean solar time. The uniform motion of the starsis equivalent to assuming that the daily rotation of the Earth is uniform. According to Audoin and Guinot(2001, 46-48), Kepler mentioned the possibility of some irregularities in the Earth’s rotation, Flamsteed wasthe first (1677) to try to detect them (without success), Maupertuis (1752) wondered whether there mightbe some irregularities and, if so, what their cause could be, and Kant suggested that “there could be slowingdown effect due to dissipation of energy in tidal movements of the oceans. He was right, but the idea wasonly confirmed by observation two centuries later.” Laplace (1825) rejected the possibility out of hand. It wasnot until the twentieth century that tiny irregularities in the rotation of the Earth were confirmed. See Smith2014, 302-07.
-  For more details, see Audoin and Guinot 2001, 40ff.