A farmer's son, William developed an early interest in mathematics which was fostered by those who recognised the boy's abilities. Such was his aptitude that Bayly came to the attention of Nevil Maskelyne (1732–1811), the Astronomer Royal, and was duly appointed as an assistant at the Royal Observatory, Greenwich. Bayly would serve as astronomer on Cook's second and third (1776–1779) voyages, otherwise fulfilling his duties at the observatory until 1785. During this latter period, Bayly was involved in writing the scientific account of Cook's third voyage. On leaving the observatory, Bayly took up the post of Headmaster at the Royal Naval Academy at Portsmouth, a position he held for 22 years before retiring aged 70. No portrait of Bayly is known to exist (Orchiston, 2016, pp. 152–153).

Like Bayly, William Wales, Fig. 1, showed an ability in mathematics from an early age and, in 1766, was commissioned by the Astronomer Royal to carry out the computations for the first edition of The Nautical Almanac. After returning from Cook's second voyage Wales accepted the position of Master of the Royal Mathematical School and also undertook, with Bayly, the task of publishing the scientific observations made during the voyage. Later still, Wales completed the scientific account of Cook's first voyage which he published 17 years after that voyage had ended. In 1765, Wales married Mary, the sister of Charles Green, the astronomer on Cook's first voyage, and his accomplishments were recognised when he was elected Fellow of the Royal Society in 1776. At the time of his death, Wales was Secretary of the Board of Longitude (Orchiston, 2016, pp. 154–155).

In the extensive introduction to their account of the astronomical observations that they made during the voyage (Wales and Bayly, 1777), the authors describe the use of the various items of equipment with which they had been provided, but tide gauges are conspicuously absent from the inventory. It would seem that Bayly and Wales were each responsible for supplying or constructing their own tide gauge device.

Bayly used a narrow glass tube that functioned as a stilling well during his first visit (May 1773) to Queen Charlotte's Sound, with the zero being set near low water, but there were times when he recorded negative tide heights. Upon his return in December 1773, Bayly deployed a pair of wooden posts that served as tide boards from which visual readings could be made. Bayly read the water level on his gauges to a precision of 0.25 in. (∼6 mm).

Wales fabricated a square wooden tube into a stilling well at Dusky Bay, and he assembled a simple dual-post arrangement half a year later at Queen Charlotte's Sound, as his tube had, in the meantime, been stolen by natives at Tahiti (Wales and Bayly, 1777, p. 56). Wales set zero for his gauges at both locations high so that all his water levels were measured below the reference mark, normally to the nearest inch (only on rare occasions to half that interval).

More detailed descriptions of these tide gauges are included in the discussion of each set of tide observations (see Sect. 6.1–6.4).

The following discussions will show that the astronomers' measurements at high water were more comprehensive than those of low water, with the latter being rarely more than a single observation. In addition to timing and measuring the highest water level, equal numbers of tide gauge readings were made before and after the maximum level. The clock time was recorded at every observation as well as the apparent time

Apparent or “sun-dial” time differs from local mean time by up to approximately a quarter of an hour due to the “equation of time” (Hughes et al., 1989).

Bayly's observations were often made at 10 min intervals and were twice as numerous as those made by Wales. Figure 2 shows a typical sequence of sea level measurements made by Wales at Dusky Bay (in this case high tide on 17 April 1773). Here, we see his gauge readings when the water level was 7, 10, and 14 in. below, before and after, high water; unlike Bayly, Wales did not make his observations at regular time intervals. As the distances to the water level were measured down from the top of the tube, the least distance is associated with high water. Clock time is given for each measurement, and apparent time is included for the time of high water.

As mentioned previously, Bayly and Wales paid far less attention to low water than they did to high water. Indeed, low-water observations normally consisted of a single reading of the gauge, and many by Wales comprised a height only, without an associated clock time. Bayly appears to have been more conscientious, often recording the period that the water level remained steady during his low-water observations; an example of such a comment is shown in Fig. 3: “The Water sunk 21/4 Inches below nothing on the scale & stood from 6h.20′ to 6h.50′ & then began to rise”. (In the preceding quote, “21/4” denotes 2.25 in.)

Whilst there appears to be no reason to doubt the veracity of the observations that Bayly and Wales recorded in their handwritten records (Bayly, 1774b; Wales, 1775b) and subsequently published in Wales and Bayly (1777), these surely cannot be the only gauge readings that they made to determine the time and height of these tide events. The handwritten records (as illustrated in Fig. 2) are too immaculate to have been penned at the gauge, and so it seems likely that a subset of the observations was transferred from a field book to their respective observation log books (Bayly, 1774b; Wales, 1775b). Without access to those original records, it is not possible to know how extensive their tide measurements really were, nor the criteria used to select the subset of observations that were published (and used in this paper).

Modern tide stations at Motuara Island and Many Islands, within Queen Charlotte Sound/Tōtaranui and Tamatea/Dusky Sound, respectively, are indicated by the blue squares in Fig. 4a and b, respectively, whilst the locations of the 1773 observations are shown by the red dots.

Harmonic analysis, using software based on that of Foreman (1977, updated 2004), of hourly sea level data recorded at Motuara Island (over 8 months during 2022) and Many Islands (44 d during 2017) produced a set of tidal constituents (harmonic constants) for each island.

Foreman's software can also be used to combine the amplitudes and phase lags of a location's tidal constituents to create tide predictions for a specified period of time for that place.

Normally, the times of such tide predictions would be in terms of a standard time meridian – for New Zealand today this is 180∘ E, New Zealand Standard Time (NZST). Obviously, Bayly and Wales were not operating in today's time zone regime; however, their recorded clock and apparent times can be converted to local mean time (see Sect. 5.1). To generate tide predictions in terms of local mean time, the phase lags of the modern harmonic constants for Motuara Island and Many Islands were transformed from the standard meridian (180∘) to the meridians of Meretoto/Ship Cove

The location within Queen Charlotte's Sound where Bayly and Wales established their tide gauges; known to them as Ship Cove but now called Meretoto/Ship Cove.

Foreman's software has been used to harmonically combine those constants to calculate the times and heights of high and low waters, as well as heights at 10 min intervals for plotting purposes (see Figs. 5, 6, 7, 8, 10, 11, and 12), throughout 1773 at Ship Cove and Pickersgill Harbour.

In order to compare the observations made by Bayly and Wales to the predicted data, both datasets must be expressed in terms of the same time system.

As mentioned above (Sect. 3.2 and 3.3), a clock time was recorded for almost every tide gauge observation, with apparent time included whenever the time of high or low water was recorded with a clock time. Whilst the reliability and accuracy of clocks were improving during the 18th century, they were not yet perfect. For this reason, the astronomers were required to attend to the watches and to compare them against apparent time derived from morning and afternoon observations of the Sun as well as against one another. The time kept by the clocks entrusted to Bayly and Wales (they had two time-keepers each) could differ from true time by 1–2 h; thus, the astronomical determinations of time (apparent time) were needed to find the offsets and rates of the clocks.

The rate of apparent time is not uniform due to the varying speed of the Earth in its elliptical orbit around the Sun; thus, this irregularity, known as the “equation of time”, needs to be corrected for. Accordingly, the apparent times of the high and low waters published in Wales and Bayly (1777) were transformed to local mean time by applying the equation of time obtained from an online calculator (PlanetCalc, 2022). By differencing each local mean time derived in this manner and its associated clock time, the clock error at that local mean time could be found. The local mean time of each tide gauge observation could then be calculated by applying the clock error, as well as any correction for clock drift, to the observation's clock time.

The 1773 observations could be compared to the predicted tides as the latter were also expressed in local mean time, as described in Sect. 4.2.

As the historic and predicted sea level heights have no common vertical reference, it is not possible to relate the heights of the observed and predicted tides in an absolute way. Therefore, in order to investigate sea level heights, we must turn to comparing tidal ranges as calculated by differencing the heights of sequential high and low waters.

When introducing the gauges erected by Bayly and Wales (Sect. 3.1), it was noted that the zero point of their measurements was seldom near the lowest level to which the water fell.

When Bayly returned to Ship Cove in December 1773, he set up two poles that were offset vertically such that the zero of the higher pole (for measuring high water) was above the pole used for low-water observations. This offset must be added to the high-water readings to bring them into terms with the low-water measurement values.

All of Wales's measurements were made downwards; therefore, these data have been inverted and expressed relative to an arbitrary zero below low water to avoid negative values.

After the historic measurements had been converted to “upward” values, an arbitrary offset was applied to each series of observations to enable the observed and predicted datasets to be plotted together in Figs. 5, 7, 8, 11, and 12.

Following the Resolution's arrival in Dusky Bay on 26 March 1773, Wales set about first establishing his observatory, before turning his attention to the tides.

In his journal, Wales describes the construction of his tide gauge as well as the modifications required before measurements could commence. Wales learns a valuable lesson when he finds the gauge to be too short after incorrectly judging the rise and fall of the tide against the shoreline (Wales, 1774, p. 127):  

3 April. Fixed up a Thermometer & Barometer, and made a machine for trying the Tydes [sic]: It consists of a long square Tube whose internal side is about 3 inches: A square float is fitted to this Tube & fixed to the end of a long slender Rod which is divided into feet and Inches from the float upwards. I propose to put down this rod into the Tube until the float just touches the water & then mark the feet & Inches on the Rod which are even with the top of the Tube: As the water is addmitted [sic] into the Tube only by a small aperture at the bottom the rise and fall of the water occassioned [sic] by the surf will be inconsiderable or at least much lessened.

Wales observed at least one tide every day for the following 3 weeks, measuring either water level height and time before, at and after high or low water, or the height alone at high or low water. In total, he recorded 27 high and 24 low waters, as shown in Fig. 5.

Table A1 lists the observed and predicted times and heights for each high and low water observed by Wales at Pickersgill Harbour as well as the calculated tidal ranges. The table also gives the differences between observed and predicted times and tidal ranges. Of the 32 time differences, only 2 exceed 30 min (both being 39 min), whilst 4 range differences are greater than 20 cm (the largest being 29 cm). These results are testament to the quality of Wales's observations.

In his Resolution log book, Wales notes that “The time of High water on the Full and Change Days was at 10H57′; and they both agreed; but the water rose 8 feet at the former, and only 5 feet 8 inches at the Latter: I cannot account for this difference; but am certain it is not owing to any error in the Observations” (Wales, 1775a, p. 38).

In his journal, Cook commented that the difference “is a little extraordinary and probably was occasioned at this time by some accidental cause such as Winds &ca” (Beaglehole, 1961, p. 138). (In the preceding quote, “&ca” denotes et cetera.)

Wales was right to trust his observations, as the difference in the ranges of successive spring tides is indeed a real feature of the tide in Dusky Sound, as illustrated in Fig. 6.

Whilst the difference between periods of spring and neap tides varies for each month, it was greatest in 1773 during the period that Wales carried out his tide gauge measurements. Furthermore, whilst Wales observed the greater range at full Moon, this is not always the case as can be seen from August onwards when the larger spring tides occur close to new Moon.

Of course, it would be unreasonable to expect that Wales and Cook should have understood the tidal regime in greater detail than the spring–neap cycle. Knowledge of the tides was still evolving in the late 18th century, and the reasons for their variability were yet to be properly understood (Cartwright, 1999).

In early February 1773, the Adventure, having become separated from the Resolution in the Southern Ocean, made for Queen Charlotte's Sound in accordance with the instructions that Cook had handed to Furneaux (Beaglehole, 1961, p. 683), arriving there on 7 April for what turned out to be a stay of 2 months.

A period of 1 month later, Bayly commenced his tide measurements, observing 20 high waters and 14 low waters in Ship Cove over the following 2 weeks, as shown in Fig. 7.

In the Adventure's log book, Bayly describes his tide gauge and the precision to which he was able to measure the water level. A glass tube, 4 ft (1.22 m) long and 0.7 in. (1.8 cm) in diameter, was attached to a long wood rod which was divided into 1 ft, 1 in., and 0.25 in. intervals (30.5, 2.5 and 0.6 cm, respectively) to serve as a scale. The rod was, in turn, attached to a post set firmly in the water. A bamboo cane, with a very small hole, fitted to the tube provided a narrow aperture for admitting the water into the glass tube. The tube performed admirably as a stilling well, and Bayly found that he could read the height of the water in the tube to 0.25 in. (∼6 mm), or better, and that the water level fluctuated no more than 0.1 in. (∼2.5 mm) even as the sea rose and fell by 1 ft (∼30 cm) due to wave action (Bayly, 1774b, p. 36).

Table A2 lists the observed and predicted times and heights for each high and low water observed by Bayly at Ship Cove during May 1773 as well as the calculated tidal ranges. The table also gives the differences between the observed and predicted times and tidal ranges. Whilst a significant number (41 %) of Bayly's time differences are greater than 30 min, this is not necessarily indicative of careless work. With a relatively small tidal range at this location (spring tide: ∼1.6 m; neap tide: ∼0.5 m), the change in water level within 30 min of high or low water at spring and neap astronomical tides is less than 30 and 10 mm, respectively. Estimating the true time of high or low tide under such circumstances would not be easy. Although most (18) of Bayly's tidal range differences are less than 15 cm, differences of 30–60 cm between 17 and 20 May are clearly evident in Fig. 7. These results can not be explained, as Bayly noted that the weather during this period was either “quiet and serene”, “calm and serene”, “good”, “calm”, or “fine”, with sea conditions described as being “still”, “undisturbed”, or “smooth”.

Having meanwhile sailed from Dusky Bay, the Resolution regained the company of the Adventure at Ship Cove in Queen Charlotte's Sound. Prior to both vessels departing for the South Pacific on 7 June, Cook wrote at length in his journal regarding this stopover at Ship Cove, including the following:

Mr Bayley commun[i]cated to me … High-Water at the full and change of the Moon at 9 o'Clock & the greatest rise 61/2 perpendicular; … Tides are confirmable enough to the like observations I made when I was here in Janry 1770, … (Beaglehole, 1961, p. 173).

This claim that the greatest tidal range is 6 ft 6 in. (1.98 m), referred to as “61/2” in the quoted material above, is curious. According to Bayly's observations log, the greatest range, on 8 May 1773, was 5 ft 6.6 in. (1.69 m) (Bayly, 1774b, p. 36). The predicted tide range on this date is 5 ft 4 in. (1.63 m), which is in close agreement with his measurements that day. It would seem that Cook did not misquote Bayly, as Bayly wrote “6Ft..5In perpendicular” in his Adventure log book (Bayly, 1774a, p. 63). Modern sea level data show the largest astronomical range at Motuara Island to be 5 ft 9 in. (1.76 m); Bayly's observations fall within this range, although not the conclusion in his log.

Cook had spent 3 weeks at Ship Cove during his first voyage to the Pacific on the Endeavour and found that the tides “flow 9 or 10 oClock at the full and change of the Moon and rises and falls upon a perpendicular 7 or 8 feet”. (Beaglehole, 1955, pp. 246–247). As noted in Woodworth and Rowe (2018), there is no information on how Cook made his tidal observations on the Endeavour voyage but, by whatever method he employed, Cook's range is clearly overstated. Nevertheless, he welcomed Bayly's (overstated) result as bestowing some degree of confirmation of his own.

After voyaging in the South Pacific and spending time in Tahiti, the Resolution returned to Queen Charlotte's Sound on 3 November 1773 and stayed at Ship Cove for 3 weeks. Wales commenced water level measurements on 13 November and, over the next 8 d, made detailed observations of 10 high waters and untimed water level readings of one high and 11 low waters, as shown in Fig. 8.

Table A3 lists the observed and predicted times and heights for each high and low water observed by Wales at Ship Cove during November 1773 as well as the calculated tidal ranges. The table also gives the differences between the observed and predicted times and tidal ranges. The greatest time difference is 19 min and the largest range difference is 17 cm – these results are commendable.

As mentioned earlier (Sect. 3.1), Wales's rectangular wooden tube was stolen from him in Tahiti; therefore, his “observations were made by means of two posts, divided into feet and inches, from their tops downwards” at Ship Cove (Wales and Bayly, 1777, p. 64). Wales does not explain the need for two posts and, as their tops were set level, the second one would seem to be quite unnecessary.

Wales (1775a, p. 69) states the following:

I further found that the time of High-water preceeded [sic] the moon's transit over the meridian by 3 Hours, and the greatest rise of the Water whilest I was here was 5 feet & 10 Inches; but there were evident tokens on the Beach of the Water's having risen full 2 feet more than that, since we were here before.

Wales's determination of the rise agrees with Bayly's assessment 6 months earlier and the predictions, as discussed in the previous section. However, the higher line of tokens (possibly referring to driftwood) on the beach would most likely have been driven there by wave action, not the astronomical tide. Perhaps it was evidence similar to this (i.e. driftwood) that led Cook, in 1770, to believe the rise was 7 or 8 ft. (2.13 or 2.44 m, respectively). The line of driftwood is still used today as an indicator of tidal inundation, although with due caution as the line is almost always landward of the actual high-water line.

To explore Wales's statement about the interval that high water preceded lunar transit, Table 2 lists his times of high waters prior to lunar transit, lunar transit time, and the time difference between these two events. Lunar transit times have been calculated using Gray (2020); see Table C3 in Appendix C. The average difference of 3 h 53 min is 1 h longer than Bayly found it to be.

Wales also made the following remark:

In these 8 Days the time of High-water has advanced only 4H,,44′: it ought, I conceive, to have advanced 6H,,40′: whatever may have been the Cause of this, I am certain it arises not from an error in the Observations, although circumstances did not admit of my making them with so much Accuracy as might be wished. (Wales, 1775b, p. 305).

Figure 9 shows Meretoto/Ship Cove (red dot) situated near the northern entrance to Queen Charlotte Sound/Tōtaranui, beyond which lies Cook Strait which separates New Zealand's North Island from the South Island. The sound is a drowned river valley that extends more than 30 km south-westward and connects to Tory Channel/Kura Te Au, another drowned valley, which heads eastwards and also connects to Cook Strait.

The flood and ebb of the tide through this region are affected by the complex nature of these waterways (Heath, 1977). One of the consequences of this regime is the variability in the interval between tides. Wales's expectation that the advance of high water over 8 d should have amounted to 6 h 40 min was based on the length of a lunar day being 50 min longer than a solar day.

Figure 10 shows that the daily progression of high waters throughout 1773 oscillated from 29 to 119 min because of the role of tidal constituents other than the predominant lunar semi-diurnal tide (M2); thus, the interval can be as short as 24 h 29 min or as long as 25 h 59 min. The mean and median daily advance times are 51 and 40 min, respectively. Wales's 8 d of measurements fell within a period when the daily progression was near the low end of its range (as indicated by the part of the curve coloured red) with the average over those days being less than that expected, hence his concern about the accuracy of his observations. We, however, know that the effect is real and that it was the quality of his work that detected the phenomenon which led Wales to question his abilities. Had Wales commenced observations a few days earlier, he would surely have been surprised that the daily advance was twice that expected.

On 25 November, the Resolution set off to explore higher latitudes during the summer before retreating to the tropics for much of 1774. After calling at the Marquesas Islands, Tahiti, New Hebrides, and New Caledonia, Cook brought the Resolution back to Ship Cove in October 1774 for a 3-week stopover before departing for England via Cape Horn. Although Wales carried out a considerable number of astronomical observations during this sojourn, no further tide measurements were made.

Towards the end of October 1773, the Resolution and Adventure became separated off the east coast of the North Island of New Zealand. Furneaux had great difficulty in getting to Ship Cove, the agreed rendezvous location should the ships part company (Beaglehole, 1961, p. 689), eventually arriving several days after Cook had departed. The ship's company set about refitting the Adventure, restocking water and wood, and preparing to depart for England on 23 December.

Bayly's tide observations during this stay, carried out over a period of just a few days, comprised only three high waters and their following low waters, as shown in Fig. 11. Bayly set up two posts at different heights and measured the vertical offset of their zero marks to be 4 ft 0.75 in. (1.24 m). The posts were, in effect, one gauge in two parts: one for high water, the other low water. The offset distance had to be added to the high-water readings to obtain values consistent with the low-water observations.

Bayly noted the following:

Here follow a few observations on the Tides at the bottom of Ship Cove where I was obliged to set up two posts one for to observe high water at, & the other to observe low water at, where nothing (on the Scale of my Tide Instrument) was always fixed to a notch or stroke across each post, this notch on the post at high water was just 4:03/4 above the level of the notch on the post at Low water; therefore nothing on the Scale at high water was so much above nothing on the Scale at Low water (Bayly, 1774b, p. 58).

(In the preceding quote, “4:03/4” denotes 4 ft 0.75 in.)

Table A4 lists the observed and predicted times and heights for each high and low water observed by Bayly at Ship Cove during December 1773 as well as the calculated tidal ranges. The table also gives the differences between the observed and predicted times and tidal ranges. The greatest of the six time differences is 22 min, and the three range differences are 22, 21, and 12 cm. It would not be appropriate to draw any conclusions from this small sample.

Bayly also noted the following:

The water seemed to rise about 6 Inches higher when the Moon was above the horizon than when she was below it. Being obliged to fix my posts at some distance from the Tents, rendered it unsafe to observe the night Tides (ibid.).

When he had visited Ship Cove earlier in the year, Bayly noted, alongside his water level measurements, when the Moon was above the horizon, but he did not repeat this practice with his second set of observations. Bayly's proposition that higher tides are somehow correlated with the presence of the Moon in the sky could not have arisen from this visit, as the Moon was above the horizon for all high waters that he observed, as shown in Fig. 11. Was this idea based on the observations Bayly made during his earlier time at Ship Cove? Figure 12 shows periods when he observed consecutive high waters during that visit.

Between 3 and 7 May, the tidal range was increasing from neap towards spring; hence, it follows that each high water would be higher than the preceding one, so Bayly's hypothesis could not have been based on those observations. Spring tides occurred on 8 and 9 May and, indeed, the diurnal inequality has the higher tide both days when the Moon is visible. The height differences, by Bayly's measurements, were 5.6 and 4.5 in. (14.2 and 11.4 cm, respectively).

During the latter part of his time series, Bayly's observations not only became sparser, as shown in Fig. 12b, but once again they occurred as the tide was progressing from neap towards spring. Although it appears that his premise holds true on 17 and 20 May, this is more likely due to the neap-to-spring transition than the presence of the Moon in the sky.