Latitude by two altitudes of the Sun – Douwes’ and Riddle's methods

Abstract

In approximately 1740, Cornelis Douwes presented an algorithmic method to determine the latitude when it is impossible to observe the Sun at the meridian passage. To apply Douwes’ method, it is necessary to know two altitudes of the Sun, the time elapsed between observations, the Sun's declination at the time when the greater altitude was observed and the latitude by account. Douwes’ method, originally written in Dutch, was translated and published in English by Richard Harrison in 1759. This translation made possible the dissemination of this method throughout Europe. In 1821, James Ivory proposed a new method that was independent of the latitude by account. This method was improved by Edward Riddle in 1822. Riddle's proposal was widely disseminated throughout Europe during the 19th century. In this work, our objective is to study the reliability of these two methods. For that purpose, we will apply the algorithmic methods of Douwes and Riddle to determine the latitude using real observations made during the years 2021 and 2022. The results obtained will then be compared with the GPS (Global Positioning System) latitude to assess the reliability of each method.

1. Introduction

At the end of the 15th century, to determine the latitude, sailors used a method based on the observation of the Sun at the meridian passage, that is, at its maximum altitude. However, due to meteorological conditions, it could be difficult or even impossible to observe the Sun at that precise moment. Whenever the observation was not possible, it would be necessary to wait for the following day to obtain the latitude. In practice, due to weather conditions or overcast sky at the meridian passage, it might be impossible to observe the Sun's altitude at the meridian passage for several days. Consequently, the determination of the ship's latitude would be compromised, also compromising the success of the trip.

Aware of these difficulties, several authors searched and have developed methods to determine the latitude using two altitudes of the Sun at two different times of the day. The first method found in the literature was proposed by Pedro Nunes in 1537. It was a mechanical method since all the calculations had to be done using an auxiliary instrument, a globe, designed and built by Nunes (1537).¹ In the 17th century, another mechanical method was proposed by Estancel (1658). All these mechanical methods used an auxiliary instrument to facilitate the required complex calculations that involved products and divisions of trigonometric functions. All these calculations were time-consuming and very prone to errors, also the use of complex instruments was not easily done on board. Therefore, these methods were impractical and hard to use on board of a ship where the objective would be to determine the latitude through quick and expeditious processes. The finding of the logarithmic function, in the 17th century, by John Napier, allowed the simplification of all the complex calculations necessary to determine the latitude, allowing for the appearance of algorithmic methods whose calculations were, clearly, more simple and easily done on board.

The first algorithmic method was proposed in Dutch by Cornelis Douwes, in approximately 1740.² His method requires the knowledge of two altitudes of the Sun, the time elapsed between observations, the latitude by account and the Sun's declination at the time when the greater altitude was observed. This method was disseminated in English through the translation of Harrison (1759) and can be found in texts by Norie (1835a) and Bowditch (1837), the latter two being the internationally most known navigation manuals at the time.

Douwes’ method depended on the knowledge of the latitude by account. To avoid this drawback, in 1821, James Ivory proposed a new method that was independent of the latitude by account. The process proposed by Ivory was improved and simplified by Edward Riddle in 1822. Riddle's proposal served as the basis for numerous authors in the 19th century who were looking for quick and reliable methods to determine latitude using two altitudes of the Sun. These methods are described in texts by Norie (1835a), Bowditch (1837) and Leitão (1865), among others. At the end of the 19th century, Marcq Saint-Hilaire proposed a method that is used until today. This is a completely different method and not comparable with Douwes’ and Riddle's methods, since it permits to determine the latitude and longitude simultaneously. Marcq Saint-Hilaire's method can be used at any time of the day or the night, while Douwes’ and Riddle's methods can only be used during the day since they require Sun observations.

In this work, we will study the methods proposed by Douwes and Riddle to determine the latitude by two altitudes of the Sun. Both methods will be applied to real observations made during the years 2021 and 2022 on trips in the Atlantic ocean and the results obtained will be compared with the GPS (Global Positioning System) latitude to assess the reliability of each method.

2. Latitude by two altitudes of the Sun – Douwes’ Method

To determine the latitude, the navigators used a method based on the observation of the Sun at its meridian passage. To apply this method, the Sun must be visible at that exact moment, which might not always be the case, making it impossible to determine latitude. In 1740, Cornelis Douwes presented a method to determine the latitude using two altitudes of the Sun. This is considered to be the first algorithmic method and the first method to use the logarithmic function. It is also considered the precursor of all other methods that followed. The method proposed by Douwes requires the knowledge of two altitudes of the Sun, the time elapsed between observations, the Sun's declination at the time when the greater altitude was observed and the latitude by account. The method is iterative, that is, at the end, the obtained latitude must be compared with the latitude by account, evaluating the difference. If the difference is large, the process is repeated, using the latitude obtained in the previous step as the latitude by account. This method, originally written in Dutch, was disseminated in English through the translation of Harrison (1759). In this work, our main goal is to apply Douwes’ method to real observations to assess its reliability. Even though there are several manuscripts where this method can be found, we will use the 11th edition of the book A New and Complete Epitome of Practical Navigation written by J. W. Norie in 1835.³

Prior to the explanation of Douwes’ method, we must make some notes and clarify a few concepts and notation.

First of all, we shall note that the observed Sun's altitudes have to be corrected for index error (if necessary), dip, refraction and parallax. Also, the value of the Sun's declination has to be reduced to the meridian of the ship, this is called the reduced declination. All the Sun's altitudes that appear in this work have been corrected and all the Sun's declinations are in fact reduced declinations, however, and for the sake of simplicity, we will simple say Sun's declination instead of Sun's reduced declination. The way these corrections are done is out of the scope of this paper; however, the interested reader can see a detailed explanation on how to do these corrections in Norie (1835a, p. 180).

With the appearance and dissemination of the logarithmic function, two ways of mentioning the trigonometric functions have appeared, namely the name of the function or the name of the function accompanied by the term artificial, and the name of the function accompanied by the term natural. When associated with the term natural, it is meant the value of the function itself. If the name of the function appears alone or with the term artificial, then it is meant the value of the logarithm of that function.⁴ For example, the sine of 30$^\circ$ corresponds to $\log (\sin (30^\circ ))$, while the natural sine of 30$^\circ$ corresponds to $\sin (30^\circ )$.

The values of these functions were obtained from a set of tables, namely the Table of Logarithmic Sines, Tangents and Secants and the Table of Natural Sines.⁵ To simplify the calculations, the number 10 (in some cases the number 5) has been added to all the values of the trigonometric functions that appear in the tables. This little trick had the objective to transform all negative numbers into positive numbers, which made all the calculations much easier. For example, if we search for the value of the artificial sine of $34^\circ$ in the logarithmic tables, we will find the number 9$.$747562, which is exactly the value of $\log (\sin (34^\circ ))+10$ (see Figure 1). The addition of a positive integer was common at that time and was used by several authors to simplify the calculations that had to be done.

Figure 1. Value of the artificial sine of $34^\circ$ in the logarithmic tables (Norie, 1828, p. 150)

To simplify his method and all the calculations that had to be done, Douwes introduced some auxiliary functions, namely the functions rising, half elapsed time and middle time. For all these functions, he has also created the tables with the necessary values needed for the resolution of his method.

The function rising is defined as $\mathrm {rising}(x)=1-\cos (x)$ and its values can be found in the table Logarithms for Rising. The parameter elapsed time, denoted by $t$, represents the time between observations, that is, $t=|t_2-t_1|$ where $t_1$ and $t_2$ denote the times when the two Sun's altitudes where observed. The parameter middle time, denoted by $mt$, corresponds to the value ${(t_2+t_1)}/{2}$. For these functions, the values necessary for the resolution of the method are $-\log \sin (t/2)$ and $\log (2\sin (mt))$. Therefore, the table Logarithms for Half Elapsed Time lists the values of $-\log \sin (t/2)$ and the Table for Middle Time contains the values of $\log (2\sin (mt))+5$.

Before we state all the steps of Douwes’ method, we must introduce the starting variables. Let $a_1$ and $a_2$ denote the two altitudes of the Sun, with the assumption that $a_1 >a_2$; let $t_1$ and $t_2$ denote the time of the observations of $a_1$ and $a_2$, respectively; let $\delta$ denote the Sun's declination at the time when the greater altitude was taken, that is, the Sun's declination at the instant $t_1$, and let $\varphi _a$ denote the latitude by account. We will now describe the seven steps of this method as described by Norie (1835a, p. 191). For each step, we will also include the respective mathematical formulation.

1. 1. To the log secant of the latitude by account, add the log secant of the Sun's declination; their sum, rejecting 20 from the index, call the log.ratio:

   \[ \log \sec \varphi_a + \log \sec \delta = \log (ratio) \]

2. 2. From the natural sine of the greater altitude, subtract the natural sine of the less altitude and set the logarithm of their difference under the log.ratio:

   \[ \log ( \sin (a_1)-\sin (a_2)) \]

3. 3. Take out the logarithm answering to half the elapsed time and set it likewise under the log.ratio:

   \[ -\log \sin \left( \dfrac{t}{2}\right) \]

4. 4. Add these three logarithms together and find the middle time ($mt$) corresponding to their sum, the difference between which and the half elapsed time, will be the time from noon when the greater altitude was observed:

   \[ \log (ratio) + \log ( \sin (a_1)-\sin (a_2))+ \log \sin \left( \dfrac{t}{2}\right) = \log (2 \sin (mt)) \]

5. 5. From the log.rising, answering to this time, subtract the log.ratio and the remainder will be the logarithm of a natural number ($n$); which being found and added to the natural sine of the greater altitude, their sum will be the natural co-sine of the meridian zenith distance ($mzd$):

   \[ \log \mathrm{rising}\left( \dfrac{t}{2} -mt \right) - \log (ratio)=\log(n); \qquad n+\sin(a_1) =\cos(mzd) \]

6. 6. Having found the meridian zenith distance, apply to it the declination and the result will be the latitude, $\varphi$, at the time of taking the greater altitude:

   \[ \text{If } mzd \text{ and } \delta \text{ are both North or South, then } \varphi=mzd + \delta \text{ else } \varphi=|mzd - \delta|. \]

7. 7. If the latitude thus found should differ considerably from the latitude by account, the operation is to be repeated, using the computed latitude instead of that by account, until the latitude last found agree nearly with the latitude used in the computation.

In the method listed above, it is assumed that both Sun's altitudes were taken at the same place; however, this rarely happens if the ship is navigating. Therefore,

[…] it will be necessary when the ship is making much way, and the elapsed time is considerable to correct the less altitude, in order to find what it would have been, had it been taken at the place where the greater altitude was observed. (Norie, 1835a, p. 194)

According to Douwes (Norie, 1835a, p. 194), to correct the lesser altitude, we have to proceed as follows.

1. (a) With the compass, find the number of points between the Sun's azimuth at the time when the lesser altitude was observed and the ship's course. Subtract from 16 if the value is greater than 8 points.

2. (b) Compute the distance sailed during the elapsed time.

3. (c) Entering this number of points and the distance sailed in Table I,⁶ and obtain the corresponding difference of latitude. This value is the correction value to be applied to the lesser altitude.

   1. (i) If the lesser altitude was observed in the morning, then add the correction if the angle determined above is less than 8 points, otherwise subtract the correction.

   2. (ii) If the lesser altitude was observed in the afternoon, then subtract the correction if the angle determined above is less than 8 points, otherwise add the correction.

In Figure 2, we can see an application of this method. In the first block, we have the procedure to determine the Sun's true altitudes. In the second block, we have the first iteration of the method and finally, in the last block, a second iteration of the method.

Figure 2. Example of Douwes’ method (Norie, 1828, p. 191)

To conclude the description of the method, there are some remarks concerning the times of the observations.

As the above method is only an approximation to the truth, it must be used under the following restrictions. (Norie, 1835a, p. 192)

According to Douwes, the hours of the Sun observations must take into account the following restrictions (Norie, 1835a, p. 192):

1. (a) observations must be carried out between 9 h and 15 h;

2. (b) if both observations are in the morning or both in the afternoon, the time elapsed between observations must be greater than the time elapsed between noon and the time of greatest altitude;

3. (c) if the observations are one in the morning and one in the afternoon, the time elapsed between observations must not exceed 4 h 30;

4. (d) in all cases, the closer the greater altitude is to noon, the better.

In Section 4, we will implement Douwes’ method in an Excel spreadsheet and use it to determine the latitude using real observations of the Sun. The values of the latitude obtained will be compared with the GPS latitude to determine the error and evaluate the reliability of the method.

3. Latitude by two altitudes of the Sun – Ivory and Riddle's method

The method proposed by Douwes to determine the latitude depended on the knowledge of the latitude by account. This was considered to be an inconvenience since it required the regular estimation of the latitude, which was a time-consuming process. To avoid this inconvenience, in 1821, James Ivory (Ivory, 1821) proposed a new method that was independent of the latitude by account.

This method of finding the Latitude by two Altitudes of the Sun, which is much simpler and more general than the former, and independent of the Latitude by Account, was proposed by Mr. James Ivory, who has given an ingenious Solution of it in the Philosophical Magazine for August 1821. Mr. Riddle, of the Royal Naval Asylum, Greenwich, has since considerably improved Mr. I.'s Solution, and given a Rule, similar to the above, in the same work for September 1822. (Norie, 1835a, p. 198)

In 1822, Edward Riddle published a paper in the Philosophical Magazine (Riddle, 1822) where he proposes some simplifications on Ivory's method. The simplifications proposed by Riddle transforms Ivory's method into a simpler and more accessible procedure.

By a very simple trigonometrical transformation, he adapted Ivory's solution to logarithmic computation, and gave to it that practical working form now in use. (Report 1855, p. 113)

As before, we shall use as reference the 11th edition of the book A New and Complete Epitome of Practical Navigation written by J. W Norie in 1835. This method considers the true altitudes of the Sun, that is, the Sun's altitudes corrected for index error (if necessary), dip, refraction and parallax. As it was noted before, the interested reader can see all the rules to perform this correction in Norie (1835a, p. 180). The lesser altitude has to be corrected for the time when the greater altitude was taken. This procedure is done as it was explained before in Douwes’ method.

Prior to the description of the steps of Riddle's method, we have to introduce the initial variables. Let $t_1$ and $t_2$ represent the time, in hours, when the greater and lesser altitudes were observed, respectively. The Sun's greater and lesser altitudes are denoted by the variables $a_1$ and $a_2$, respectively, and let $\delta$ denote the Sun's declination for the time $t_1$, that is, for the time when the greater altitude was taken. The elapsed time, denoted by $t$, represents the time between observations, that is, $t=|t_1-t_2|$. Finally, we shall observe that all the values of the trigonometric functions mentioned in the method are actually the values of their logarithms (base 10) and for the sake of simplicity, we shall omit the $\log$ function. For example, when it is mentioned, the secant of the declination, mathematically speaking, it is meant the $\log \sec (\delta )$. We will now describe the eight steps of this method as described by Norie (1835a, p. 198). For each step, we will also include the respective mathematical formulation.

1. 1. Add together the true altitudes (found as before) and take half their sum; subtract the lesser altitude from the greater and take half their difference:

   \[ \dfrac{a_1+a_2}{2}; \quad \dfrac{a_1-a_2}{2} \]

2. 2. Find the interval between the times of observing the two altitudes, called elapsed time, take half the elapsed time and reduce it to degrees:⁷

   \[ \dfrac{t}{2}=\dfrac{|t_1-t_2|}{2} * 15 \]

3. 3. Add together the co-secant of half the elapsed time and the secant of the declination; their sum will be the co-secant of arc first:

   \[ \csc \left( \dfrac{t}{2} \right) + \sec (\delta) = \csc (arc \ first) \]

4. 4. Add together the co-secant of arc first, the co-sine of half the sum of the altitudes and the sine of half their difference: the sum of these logarithms will be the sine of arc second:

   \[ \csc(arc \ first) + \cos \left (\dfrac{a_1+a_2}{2} \right) + \sin \left (\dfrac{a_1-a_2}{2} \right) = \sin (arc\ second) \]

5. 5. Add together the secant of arc first, the sine of half the sum of the altitudes, the co-sine of half their difference and the secant of arc second; their sum will be the co-sine of arc third:

   \[ \sec(arc \ first) + \sin \left (\dfrac{a_1+a_2}{2} \right) + \cos \left (\dfrac{a_1-a_2}{2} \right) +\sec (arc\ second)= \cos (arc\ third) \]

6. 6. Add together the secant of arc first and the sine of the declination; their sum will be the co-sine of arc fourth, when latitude and declination are of the same name; but when they are of contrary names, take the supplement for arc fourth:⁸

   \[ \sec(arc \ first) + \sin (\delta) = \cos (arc\ fourth) \]

7. 7. Take the sum or difference of arcs third and fourth, for arc fifth (see Note):

   \[ arc \ third \pm arc \ fourth = arc\ fifth \]

8. 8. Add together the secants of arc second and arc fifth; their sum will be the co-secant of the Latitude, denoted by $\varphi$:

   \[ \sec(arc \ second) + \sec (arc \ fifth) = \csc (\varphi) \]

9. 9. Note: When the sum of arcs third and fourth is equal to or greater than $90^\circ$, their difference is always arc fifth; but when their sum is less than $90^\circ$, it may doubtful whether their sum or difference ought to be taken for arc fifth. Do both cases and one of the results must certainly be the required latitude, and the latitude by account will generally be sufficient to determine which of them ought to be taken.

As in the previous method, there are some remarks concerning the times of the observations.

In this method the observations should, if possible, be taken under the same limitations as directed in the former. (Norie, 1835a, p. 198)

In Figure 3, we can see an application of this method. In the first block, it is determined the Sun's reduced declination and the Sun's true altitudes, and in the last part, we have the various steps of the algorithm to determine the latitude.

Figure 3. Example of Riddle's method (Norie, 1835b, 18th ed., p. 201)

Riddle's method was implemented in an Excel spreadsheet and used with real observations of the Sun to determine the latitude. The value of the latitude obtained was compared with the GPS latitude to evaluate the error and assess the reliability of the method. A complete and detailed analysis of the results will be presented in Section 4.

4. Real world application of the methods

In this section, our objective is to see how the methods proposed by Douwes and Riddle behave in practice. For that purpose, we will apply both methods to determine the latitude using real observations and the values obtained will be compared with the GPS latitude.

In 2021, between the 5th and the 23rd of August, during a round trip on the tall ship Sagres between Lisbon and Azores, we collected 41 Sun altitudes. The altitudes were taken using a sextant and were corrected for index error, dip, refraction and parallax. The times of the observations were also registered as well as the GPS position. The values of the Sun's declination for the times of the observations were obtained from the Nautical Almanac for 2021 (Nautical Almanac, 2020).

This procedure was repeated in 2022 on a trip on the tall ship Sagres from Lisbon to Brazil. Between the 27th of July and the 5th of August, we collected 34 Sun altitudes, the times of the observations and the GPS position. The Sun's declinations were obtained from the Nautical Almanac for 2022 (Nautical Almanac, 2021).

Recall that to apply both methods, we need to use a pair of altitudes within the same day. Thus, for each day of observations, we built all possible pairs making a total of 129 pairs of observations, 61 pairs in 2021 and 68 pairs in 2022. For each of these pairs, the lesser altitude was corrected for the instant when the greater altitude was observed using the procedure described in Section 2.

Both methods were implemented in an Excel spreadsheet. For each pair, the latitude was determined using Douwes’ and Riddle's methods and the values obtained were compared with the real GPS latitude. Finally, a careful analysis of the results was carried out to assess the reliability of the proposed methods. We will also analyse the behaviour of the two algorithms when we travel in different directions. In 2021, the ship travelled East/West, where the latitude varies slowly with time, while in 2022, the ship sailed South, where the variation in latitude is more significant over time.

4.1 Douwes’ method

In this section, we will present all the steps needed to implement Douwes’ method in an Excel spreadsheet. Before we explain each step, we need to define the initial variables and make some general remarks. For practical reasons, all times and angular values were converted to a decimal format. The decimal angles where later converted to radians since the arguments of all trigonometric functions in Excel are in radians.

Let $a_1$ and $a_2$ denote the two altitudes of the Sun with the assumption that $a_1>a_2$. Recall that $a_2$ denotes the corrected value, that is, the value of the lesser altitude corrected to the place and time where the greater altitude was observed. This correction was done using the procedure described in Section 2. Let $t_1$ and $t_2$ denote the observation times of $a_1$ and $a_2$, respectively. Let $\delta$ denote the Sun's declination at the time $t_1$ and finally, let $\varphi _a$ denote the latitude by account, $\varphi$ the latitude determined by algorithm and $\varphi _{GPS}$ the GPS latitude.

The first thing that has to be done is to determine the time between observations, called the elapsed time and denoted by $t$. Since all the times were converted to a decimal value, the value of $t$ is easily obtained by the formula $t=|t_1-t_2|$. Then, we need to convert its value to degrees. This is done by multiplying the decimal value by 15, since 1 h corresponds to $15^\circ$. For example, if the elapsed time is 1 h and 30 min, then in decimal format, we have $t=1.5$ h or $t=22.5^\circ$. Recall that $\mathrm {rising}(x)=1-\cos (x)$.

We will now list all the steps that we have implemented in the Excel spreadsheet.

• Step 1  $\log (ratio)= \log \sec \varphi _a + \log \sec \delta$.

• Step 2  $x=\log ( \sin (a_1)-\sin (a_2))$.

• Step 3  $y= |\log \sin (t/2) |$.

• Step 4  $mt = \arcsin \left (\frac {10^{\log (ratio)+x+y}}{2} \right )$.

• Step 5  $mdz= \arccos ( 10^{\log (1-\cos ( t/2 -mt) ) - \log (ratio)} +\sin (a_1))$.

• Step 6  $\text {If } mzd \text { and } \delta \text { are both North or South, then } \varphi =mzd + \delta \text { else } \varphi =|mzd - \delta |$.

• Step 7  If $|\varphi -\varphi _{GPS}|>20'$, repeat the process with $\varphi _a=\varphi$.

Observe that Douwes’ method determines the latitude at the time when the Sun's greater altitude was observed, that is, at the time $t_1$. Furthermore, the method uses the latitude by account as a starting value. Nowadays, due to the evolution of the navigation methods, the latitude by account is no longer computed during the trip. To overcome this difficulty and be able to implement Douwes’ method, we have taken the latitude by account to be the GPS latitude for the time $t_1$, that is, we have set $\varphi _a=\varphi _{GPS}$. However, considering that latitude by account would seldom be equal to the real latitude, we decided to analyse the behaviour of the algorithm when the latitude by account is different from the true latitude. For this purpose, we have randomly added an error term to the GPS latitude. For each pair, we have selected a random number between $-$0$.$5 and 0$.$5 that was added to the value of the GPS latitude in decimal degrees. Let us denote the obtained value by $\varphi _{error}$. We have run Douwes’ algorithm with these two options for the latitude by account separately for the years 2021 and 2022.

For the year 2021, taking $\varphi _a=\varphi _{GPS}$ and after one iteration, we obtained 59 pairs with an error term less than 20 miles, this represents 97% of the total number of pairs considered. There are only two pairs with errors greater than 20 miles, one pair has an error of 20$.$3 miles, while the other has an error of 38$.$3 miles. Although these results are extremely good, we have performed a second iteration of the algorithm for these two pairs. After two iterations of the algorithm, all pairs have an error less than 20 miles, which shows that the algorithm is reliable.

In our second case, we have run the algorithm using $\varphi _a=\varphi _{error}$. After one iteration, only three pairs had an error greater than 20 miles, the greatest error being 34 miles. After two iterations of the algorithm, all pairs had an error less that 20 miles. A complete summary of these results can be found in Table 1.

Table 1. Results obtained by Douwes’ method for the year 2021

For the year 2022, we have run Douwes’ algorithm by first considering that $\varphi _a=\varphi _{GPS}$. In this case and after one iteration, we obtained 63 pairs with an error term less than 20 miles, which represents 93% of the total number of pairs considered. For the remaining five pairs, we have three pairs with errors between 20 and 40 miles, one with an error of 53$.$4 miles and one with an error of $1^\circ 49'$. For the latter, the reason why we have such a greater error has to do with measurement errors. In fact, we have compared the registered Sun's altitudes (see Table 2) with the Sun's calculated altitudes⁹ and we found that the true altitude at the time $t_1$ is 66$.$33, while the true altitude at the time $t_2$ is 48$.$39, that is, $a_1$ has an error of 11$.$5 min, while $a_2$ has an error of 2$.$5 min. This measurement error is the reason why the latitude determined by the algorithm has such a great error.

Table 2. Pair with large error in Douwes’ method for the year 2022

We have performed a second iteration of the algorithm for the four pairs with errors between 20 and 60 miles. After the second iteration, we obtained two pairs with errors less than 20 miles and two pairs with errors of 29$.$9 and 43$.$8 miles. Therefore, after two iterations of the algorithm, we have obtained good latitude values for 96% of the pairs considered.

In a second analysis of the algorithm, we wanted to see the performance of the algorithm when we consider $\varphi _a=\varphi _{error}$. In this case, and after the first iteration, we have 64 pairs with an error less than 20 miles. As before, we have one pair with an error of $1^\circ 57'$ which corresponds to the pair with measurement errors. The remaining three pairs have errors of 33$.$1, 46$.$4 and 58$.$1 miles. We have performed a second iteration of the algorithm for these three pairs and have obtained one more pair with an error less than 20 miles. Overall, and after two iterations of the algorithm, we have 65 pairs for which the calculated latitude has an error less than 20 miles when compared to the GPS value, i.e. the algorithm has a success rate of 96%. A complete summary of these results can be found in Table 3.

Table 3. Results obtained by Douwes’ method for the year 2022

Finally, we can easily observe that Douwes’ method produced slightly better results in 2021 than in 2022. One reason for this might be the fact that in 2021, the ship sailed from Lisbon to Azores and back to Lisbon, in East/West courses, where the latitude varies slowly with time, while in 2022, the ship sailed from Lisbon to Brazil, that is, mainly South or Southwest courses in which the variation of latitude with time is more significant. However, the results are extremely satisfactory in both years.

4.2 Riddle's method

In this section, we will present all the steps needed to implement Riddle's method in an Excel spreadsheet and then we will analyse its performance using the real data collected in 2021 and 2022. In Riddle's method, the initial variables are the same as for Douwes’ method except for the latitude by account. We maintain the same notation used before. In summary, the Sun's altitudes are denoted by $a_1$ and $a_2$ with the assumption that $a_1>a_2$. The times of the observations of $a_1$ and $a_2$ are denoted by $t_1$ and $t_2$, respectively, and we denote by $\delta$ the Sun's declination at the time $t_1$. Let $\varphi$ denote the latitude determined by algorithm and let $\varphi _{GPS}$ denote the GPS latitude. As before, all times are in decimal format and angular values were converted first to a decimal format and then to radians.

The first step of the method calculates half of the sum of true altitudes and half their difference. Since we are using an Excel spreadsheet, we can skip this step. Therefore, the first step of the method becomes the determination of the parameter $t/2$ called half elapsed time and its conversion to degrees using the conversion rate that 1 h is equal to $15^\circ$. This is done using the formula

\[ \dfrac{t}{2}=\dfrac{|t_1-t_2|}{2}*15. \]

We will now list the procedure to determine the five arcs that are necessary to determine the latitude. A simple observation of the method shows that to determine the value of arc first, we need to evaluate the inverse function of the function co-secant. This function is not available in Excel as a built-in function; therefore, we first need to determine the mathematical expression of inverse function of the co-secant. By observing that $\csc (x)$ is by definition $1/\sin (x)$, we can easily conclude that the inverse function of $\csc (x)$ is defined by the expression $\arcsin (1/x)$. We will now list all the steps that we have implemented in the Excel spreadsheet.

1. Step 1  $1Arc= \arcsin ( 10^{ -\log \csc (t/2)-\log \sec (\delta )})$.

2. Step 2  $2Arc= \arcsin \left ( 10^{ \log \csc (1Arc) + \log \cos \frac {a_1+a_2}{2} +\log \sin \frac {a_1-a_2}{2}} \right )$.

3. Step 3  $3Arc=\arccos \left ( 10^{ \log \sec (1Arc) +\log \sin \frac {a_1+a_2}{2} +\log \cos \frac {a_1-a_2}{2} + \log \sec (2Arc) } \right )$.

4. Step 4  $4Arc = \arccos ( 10^{\log \sec (1Arc) + \log \sin (\delta ) } )$.

5. Step 5  In this step, we will determine the value of the fifth arc, denoted by $5Arc$.

   1. (i) If $2Arc+4Arc \geq 90^\circ$, then $5Arc= 4Arc-3Arc$.

   2. (ii) If $2Arc+4Arc < 90^\circ$, then $5Arc_1= 4Arc-3Arc$ or $5Arc_2= 4Arc+3Arc$.

6. Step 6  In this last step, we determine the latitude, denoted by $\varphi$. When case (ii) happens, we will use the two possible values for $5Arc$ and determine two options for the latitude. The final value for $\varphi$ is the one that is closer to the real latitude:

   \[ \varphi= \arcsin ( 10^{ - \log \sec(2Arc) - \log \sec (5Arc)} ) \]

Recall that the latitude obtained by Riddle's method is the latitude at the time $t_1$, that is, at the time when the Sun's greater altitude was observed. We have run the algorithm for the years 2021 and 2022, and have compared the value of the calculated latitude, $\varphi$, with the value of the GPS latitude for the time $t_1$, denoted by $\varphi _{GPS}$.

For the year 2021, we obtained 60 pairs with an error less than 20 miles and one pair with an error of 22$.$2 miles. That is, the algorithm has a success rate of 100%.

For the year 2022, we have 60 pairs with an error less than 20 miles, five pairs with an error between 20 and 40 miles, and three pairs with errors greater than 60 miles. A complete summary of the results obtained for the years 2021 and 2022 can be found in Table 4, and a complete list of the pairs with an error greater than 60 miles can be found in the Table 5.

Table 4. Results obtained by Riddle's method for the years 2021 and 2022

Table 5. Pairs with large error in Riddle's method for the year 2022

As we have seen before in Douwes’ method, the pair 2253 has a measurement error in the Sun's altitudes. The error in the value of $a_1$ is 11$.$5 min, which is quite significant, hence the latitude obtained has an error greater than 60 min.

With regards to pairs 2237 and 2247, we verified that they do not respect the restrictions imposed by the method. In fact, Riddle's method has some restrictions to the times of observation.

If one observation be taken in the forenoon, and the other in the afternoon, the elapsed time must not exceed four hours and a half; and in all cases, the nearer the greater altitude is to the noon, the better. (Norie, 1835a, p. 192)

Both pairs have one observation in the morning and the other in the afternoon. The pair 2237 has an interval between observations equal to 7 h 25, the greater altitude was observed at 10 h 20 in the morning and the local noon was at 13 h 25. As for the pair 2247, we have an interval time of 6 h 18 and the local noon was at 13 h 39, while the greater altitude was observed at 16 h 27. Therefore, both pairs fail to comply with the restrictions, which justifies the magnitude of the error obtained.

Therefore, if we exclude the pairs 2237, 2247 and 2253, we have 60 out of 65 pairs with an error less that 20 miles, which corresponds to 92% of the admissible pairs. For the remaining five pairs, the largest error is 38$.$1 miles. These results show that Riddle's method is reliable and produces good results.

Finally, we can easily observe that Riddle's method produced better results in 2021 than in 2022. One reason for this might be the fact that in 2021, the ship sailed from Lisbon to Azores and back to Lisbon, in East/West courses, where the latitude varies slowly with time, while in 2022, the ship sailed from Lisbon to Brazil, that is, mainly South or Southwest courses in which the variation of latitude with time is more significant. However, the results are extremely satisfactory in both years.

5. Conclusion

Until the end of the 15th century, the method to determine the ship's latitude was based on the observation of the Sun at the meridian passage. This was mathematically simple and was easily done by the sailors. However, due to meteorological conditions, it could be hard or even impossible to observe the Sun at that precise moment, making it impossible to determine the ship's latitude. To overcome this difficulty, several authors have studied and developed alternative processes to determine the ship's latitude when it is impossible to observe the Sun at the meridian passage. The first algorithmic method found in the literature was published in 1740 by Cornelius Douwes. Originally written in Dutch, this method became widely known due to the English translation published by Harrison (1759). To apply Douwes’ method, it is necessary to know two altitudes of the Sun, the time elapsed between observations, the Sun's declination at the time when the greater altitude was taken and the latitude by account. The fact that this method depends on the knowledge of the latitude by account is clearly a disadvantage. Therefore, in 1821, and to overcome this difficulty, James Ivory proposed a new method that was independent of the latitude by account. Ivory's method was later improved by Edward Riddle and his proposal was widely disseminated all over Europe during the 19th century. With this work, our main objective is to see how the methods proposed by Douwes and Riddle behave in practice. For that purpose, during two trips on the tall ship Sagres, we collected 75 Sun altitudes, 41 in 2021 and 34 in 2022. The altitudes were taken using a sextant and were corrected for index error, dip, refraction and parallax. The times of the observations were also registered as well as the GPS position. The values of the Sun's declination for the times of the observations were obtained from the Nautical Almanac for the years 2021 and 2022. Each method requires a pair of altitudes within the same day. Thus, for each day of observations, we built all possible pairs making a total of 129 pairs of observations, 61 pairs in 2021 and 68 pairs in 2022. Douwes’ and Riddle's methods were implemented in an Excel spreadsheet and used to determine the latitude for the 129 pairs that we had. The values obtained where compared with the GPS latitude to determine the magnitude of the errors obtained.

The results obtained prove that both methods produce good values for the latitude. For 2021, and after one iteration of Douwes’ method, we have 95% of the pairs with a calculated latitude that has an error less than 20 miles to the GPS value, and after the second iteration, all pairs have an error less that 20 miles, which means a success rate of 100%. For 2022, from the 68 pairs considered, we have 63 with an error less than 20 miles after one iteration and 65 pairs after the second iteration, this is a success rate of 93% after one iteration and 96% after the second iteration. As for Riddle's method, the algorithm results show that the errors are less than 20 miles for 98% of the pairs obtained in 2021 and 88% of the 2022 pairs. However, for the year 2022, we have one pair with measurement errors and two pairs that do not respect the method restrictions about the times of the observations. Therefore, if we exclude these three pairs, we have 60 pairs from a total of 65 for which the calculated latitude has an error less than 20 miles, corresponding to 92% of the admissible pairs and the largest error obtained is 38$.$1 miles. Therefore, these results lead us to conclude that both algorithms behave extremely well in practice. Furthermore, we can also observe that we obtained better results for both algorithms in 2021 than in 2022. We are aware that the set of data collected in each year is relatively small, so it may be difficult to draw conclusions. However, one reason for this fact might be the difference in ship's course; in fact in 2021, the ship sailed from Lisbon to Azores and back to Lisbon, that is, it sailed in East/West courses, where the latitude varies slowly with time, while in 2022, the ship sailed from Lisbon to Brazil, that is, mainly South or Southwest courses in which the variation of latitude with time is more significant.

Finally, we would like to observe that, to our knowledge, the methods studied here did not have much practical use at sea despite having been widely disseminated internationally. One reason for this might be the fact that both methods are mathematically complex, require a huge amount of calculations and the use of several logarithmic tables. Furthermore, the method of calculating latitude through the meridian passage of the sun was far more simple making it the preferred choice of sailors even in the case when the weather conditions were adverse and they had to wait several days to determine the latitude.