Maurolico, Rheticus, and the Birth of the Secant Function

In 1551, Georg Rheticus published a compact set of tables that effectively completed the set of all six trigonometric functions. However, his work was not widespread, and may not have been known to Francesco Maurolico when he published a secant table in 1558. Before the end of the century, several authors argued whether Maurolico had borrowed the notion of the secant, and his table, from Rheticus. We present Maurolico’s text on his table (named the tabula benefica in a nod to Regiomontanus’s tangent table, the tabula foecunda), as well as a translation and analysis. Finally, we demonstrate that Maurolico’s table is too accurate to have derived from Rheticus’s work, absolving him of the historical accusation.


Edition [folio 60r]
Demonstratio Tabulae Beneficae. 10 [1]AD imitationem tabulę foecundę Ioannis de Monte regio, fecimus aliam tabulam, quam Beneficam appellauimus, quod eius beneficio calculis quibusdam facilitas procuretur. Vt autem utriusque tabulę subiectum et speculatio patefiat, Esto triangulum  [20] .A. graduum . 30 projecting the umbra BC. The sinus and gnomon are supposed to have 100,000 parts. And there will be angle A, for which we find entries both in the tabula foecunda and in the tabula benefica, and also in the table of right sines. Through this entry in the tabula foecunda, line BC, which is the umbra versa, may be found. From the tabula benefica may be found line AC, the radius, namely [the line] connecting the apex of the gnomon, namely point A, with C, the endpoint of the umbra. From the table of sines may be found BD, [10] namely the sinus of angle A, whose sinus secundus is line DA, namely the sinus of angle ABD, which is the complement of angle A, seeing as that together they make a right angle.
Therefore, since the partial triangles ABD and BCD are, by the eighth [proposition] of the sixth book of Euclid, similar to one another and the total triangle ACB, on that account lines DA, AB, and CA are in continued proportion. Hence, if the square of AB is divided by line AD, the quotient will be line CA. Likewise, just as line AB is to BD, so AC [15] is to CB. Hence, if AC is multiplied by BD and the product is divided by AB (which will be done), casting off five [digits] at the right of the figure from the productas many, namely, as there are zeroes in the number 100,000-the quotient will be BC, the umbra versa. On the other hand, just as BD is to AB, so BC is to CA. Hence, if AB is multiplied by BC (which will be done), applying five zeroes to the right of number BC, and the product is divided by BD, the quotient will be line CA. Likewise, if angle [20] A is assumed to be 30°, then BD will be the sinus dimidium, while AB will be sinus totus. But just as AB is to BD, so is AC to CB. Therefore, the umbra BC will be half of the number CA in the tabula benefica. And the square of AC will be four times the square of BC and therefore the square of AC will be 4/3 the square of AB. So, therefore, if AB is assumed to be the radius [folio 60v] [1] of a sphere, then line AC will be the side of a cube circumscribed by the sphere. But this is beside the point. In addition, if angle A is assumed to be 45° (that is, half of a right [angle]) then lines AD, DB, and DE will all be equal. So, umbra BC will be equal to its gnomon AB, and AC, the number in the tabula benefica, double AD or DB, which are the sinus primus and secundus of angle A.
[5] And thus, if AB is assumed to be the radius of a circle, AC will be the side of a square described in the circle. But this is beside the point. Finally, if angle A is assumed to be 60°, then angle C will be 30° and the same size as angle ABD. Hence, AD will be half of AB. But just as BA is to AD, so is CA to AB. Therefore, AB is half of AC. So, when AB is assumed to have 100,000 parts, in that case AC will be a number in the tabula benefica with 200,000 parts.
[10] Next, if angle A is assumed to be 90° (i.e., right), then in the table of sines, the sinus BD is now the same as the sinus totus AB. For in that case line BD is united with AB. But then, both in the tabula foecunda and in the tabula benefica, the umbra versa, and thus the radius, go to infinity. This is the case because the umbra BC and the radius AC are parallel and will not intersect, even if drawn to infinity. From these things, the definition is evident enough, as is the making of the tabulae [15] foecunda and benefica: For both may be derived from the table of sines by their own particular rules. Now for the use of the tabula benefica (for calculations with the foecunda were shown previously), learn these things: Let there be in the surface of a sphere two quadrants of great circles, ABE, and ACD, inclined toward one another. And from point F, namely the pole of AC, let there descend two quadrants, FED and FBC. The angle at CDE will be right. Then, in the spherical triangle ABC, let arcs [20] AB and BC be known. From these, I will seek arc AC thus: I look up arc BC in the tabula benefica, and find the number to be multiplied (let it be G). And since the sinus secundus of arc BC is the sinus of arc FB, then, just as was shown previously, the sinus of arc FB, the sinus totus, and G will be in continued proportion. But by Menelaus' Theorem, as the sinus of FB is to the sinus totus, arc FC, so is the sinus of BE to the sinus of ED. Therefore, as the sinus of BE is to the sinus [25] of CD, so will be the sinus totus to G. On that account, I multiply G by the sinus of BE. Then, I divide the product by the sinus totus (and let five figures be cast off from the product to the right, as many as there are zeroes in the sinus totus), and I will have in the quotient the sinus of arc CD, which, when subtracted from the quadrant ACD, leaves me with arc AC, which remains. And in this way, if AB is an arc of the ecliptic beginning from a section of the equator, then from BC I can know the declination, and from AC the right [30] ascension.
Likewise, changing points, from AB, an arc of the circle of latitude bounded by star B and the equator ACD, and from BC, the declination of the given star, I may discover arc AC of the equator, which is called the difference of transit [differentia transitus] of the star through midheaven; and by using it, one can know the right ascension.
Thus, likewise necessarily changing the proposed points, from the rising latitude [latitudo ortus] AB and given the declination BC of the star supposed to be placed in point B, I can find [35] arc AC, the ascensional difference of the same star. And continuing with other inquiries [I can find] those things that can render the description of that arc.
If, however, in the same spherical triangle ABC, the angle A is given, that is, arc DE and arc BC, then from these arc AB can be known thus. With arc FE, I go to the tabula benefica, and the number to be multiplied, G, will be found there. And since the sinus secundus of arc FE is the sinus of arc ED, just as before, the sinus of arc ED, [40] the sinus totus, and G will be in continued proportion. But, as Menelaus shows, the sinus of BC is to the sinus of AB as the sinus totus is to G. For this reason, I multiply G by the sinus of BC and divide the product by the sinus totus (which should be cast off by five figures from the right) and the result will be the sinus of arc AB, which was sought. By such calculation, given arcs BC (the declination) and ED (the maximum declination), arc AB of the ecliptic is found, [45] which corresponds with AC, the arc of ascension. And I can do similar things for other similar questions. For example, if, given angle A, the complement of the latitude of the region, and arc BC, the declination of a star located in point B, I want to obtain AB, the arc of the horizon between the equator and the star from the east, which is called the rising latitude, then I can do just as was said.
And if the two arcs AC and BC are given, then from those I can find arc AB. Indeed, arcs CD and FB, which are the complements of the givens, are known, and [50] from these, by the rule of declination, arc BE is given, and its complement AB, which was sought.
Finally, if having been given arcs AB and BC, arc ED is sought, then I will go into the tabula benefica with arc BE, and let the number found be G. And because the sinus secundus of BE is the sinus of AB, on that account, as earlier, the sinus of AB, the sinus totus, and G will be in continued proportion. But as Menelaus shows, sinus AB is to the sinus totus as [55] sinus BC is to sinus ED. Therefore as sinus BC is to sinus [folio 61r] [1] ED, the sinus totus is to G. Therefore, I multiply G by the sinus of BC and divide the produce by the sinus totus (which should be cast off by five figures, or as many zeroes as there are in the number of the sinus totus) and I have as the quotient the sinus of arc ED, which was sought. Thus, in this way, from AB, an arc of the ecliptic, and its declination, BC, I can obtain the maximum [5] declination DE.
Indeed, from this it is obvious that everything that Regiomontanus, with a squeamish [fastidium] evasion of division, worked out by multiplication in the tabula foecunda can be computed with our benefica without difficulty. But below we will set out a practical example of these sorts of rules with the tables themselves. We considered these things while we were with Lord Giovanni Ventimiglia, Marquis of Geraci, working on the Apollonian arc [10] during the month of August, 1550. 11 Now we will join to this certain older works considering the same theory of the spheres.

Commentary
We reproduce the diagrams as they appear in the text, capitalizing the letters for ease of use. Capitalized trigonometric functions ("Sin" rather than "sin") indicate the use of a base circle of 100,000.
[f. 60r lines 1-11] The standard functions are defined according to the line segments in Figure 2. A is the given angle; the unit length AB is chosen to be 100,000. Then BC is the tangent (tabula foecunda) and AC is the secant (tabula benefica), both 100,000 times the modern functions. Maurolico notes that BC is also equal to AC A sin and that [lines 14ff] After a brief remark that both the tangent and secant tables may be computed from a sine table, Maurolico transitions to spherical trigonometry. Figure 4 (drawn twice in the text) serves for the rest of the passage. All four arcs are 90° long. The spherical trigonometry is divided into four sections, each finding some element of right triangle ABC given two other elements (as well as the right angle at C), followed by applications in spherical astronomy.    [lines [28][29][30][31][32][33][34][35] Three astronomical applications of (a) are given: • • If ABE is the ecliptic and ACD the celestial equator, then AB is the longitude λ of B, and BC is its declination δ. The method in (a) may be used to find AC, the right ascension α of B. • • If AB is an arc on the circle of latitude 13 to star B, and ACD is the celestial equator (so that BC is the star's declination), then (a) may be used to find AC, the "difference of transit." 14 • • If AB is the arc from the east point on the horizon to star B and ACD is the celestial equator, then (a) may be used to find AC, the ascensional difference (the difference between the star's right ascension and oblique ascension).
. This method is equivalent to the modern identity sin sin sin a A c = .
[lines 43-48] Two astronomical applications of (b) are given: • • If ABE is the ecliptic and ACD the celestial equator, then A ED = =ε , the obliquity of the ecliptic ("maximum declination"). If B's declination δ = BC is given, then (b) may be used to find the ecliptic longitude AB. [lines 5-10] Maurolico concludes that his tabula benefica is as useful as Regiomontanus's tabula fecunda, provided that one is not "squeamish" about the divisions required by the various formulas. Divisions were a real issue for astronomers: they took more effort to perform than multiplications, and are more prone to error. As his examples in this text attest, Maurolico himself does not seem concerned about them. His comment that he will set out a "practical example" of the tables likely refers to the canons following the tables (folios 67 and 68), in which he solves several astronomical problems using both the tabula fecunda and the tabula benefica.
Finally, it is worth noting that Maurolico says he worked on this in August 1550, the year before Rheticus's Canon doctrinae triangulorum was published. One wonders whether he did so to head off (in the end, unsuccessfully) a priority dispute with Rheticus. In fact, a manuscript exists containing early notes of Maurolico's work on the tabula benefica, though not the table itself. 15

The controversy over the origin of Maurolico's secant table
Maurolico opens the Demonstratio tabulae beneficae with an explicit acknowledgment of his debt to Regiomontanus and his tabula fecunda, but does not mention Rheticus. Twenty-five years later, Thomas Fincke published one of the most popular and influential trigonometry texts prior to the invention of logarithms, the 1583 Geometria rotundi. 17 This book was responsible for several innovations, including the introduction of the words "tangent" and "secant" and the first appearance of the abbreviations "sin," "tan," and "sec." In it he comments on Maurolico's work: "Maurolico published the canon of Rheticus, with a few changes, in the Messinese edition of Menelaus, also changing the name: he called it the [tabula] benefica, as line OI, or the number it defines, may be called beneficial [benefica]. And line OI is no more beneficial than the line AI is fruitful [foecunda]." 18 Fincke's judgment in favor of Rheticus may have been influenced by Erasmus Reinhold, whose tangent tables Fincke followed in constructing his own. 19 Reinhold had not included secant tables in his Primus liber tabularum directionum (1554), but explains how to calculate them in his description of his table of tangents. 20 He also notes that he does not need to go into more detail because this is readily available in Rheticus's canon. 21 Fincke was certainly aware of this statement; he takes a demonstration, with attribution, from the section of the Primus liber in which Reinhold mentions Rheticus's tables. 22 This provides some context for Fincke's claim about Maurolico's copying. Reinhold "It is true that all mathematicians report having learned this [the structure outlined in Figure 1 and used by Viète] from the said Palatine author [i.e., Rheticus] in his works, where he first introduced the use of the secant or hypotenuse, and widened the use of the tangent (which was invented by Regiomontanus), although Francesco Maurolico, not least of the mathematicians of the previous century, may also seem to have discovered the secant, when in a certain one of his additions to the Elements of Theodosius, published at Messina in the year 1558, he constructed a table of secants, which he called benefica. Nor is it that we should suspect that the latter took anything from the former, because the methods of each are quite different, and the great canon was commended to Leipzig by [Rheticus] in about 12 pages in 1551. On account of the scarcity of the work, it would not have been able to turn up in Messina in Maurolico's hands; on the contrary, it was only offered to us by chance a few years ago." 23 The rarity of Rheticus's 1551 tables was due to the fact that they had been placed on the Catholic church's Index expurgatorius, surely one of the most unusual books on that list. 24 Magini notes that Maurolico's style differs from the approach advocated by Rheticus; indeed, Maurolico is much closer to Regiomontanus.
A couple of decades later John Wedderburn echoed Magini's view in his Quatuor problematum confutatio: "Francesco Maurolico, a Sicilian, thought himself the first to invent secant tables, although a little earlier, the Palatine had constructed the same [tables] in Germany; neither took anything from the other, as you can see from the extremely clear evidence of Giovanni Antonio Magini in the most perfect work on the primum mobile." 25 Wedderburn's use of Maurolico as an example was not an accident. The Quatuor problematum confutatio was written as a response to an attack on Galileo's Sidereus nuncius by Martin Horky, Magini's secretary. 26 Horky had gone beyond the norms of polite discourse, and Magini had apologized to Galileo privately while disassociating himself from his young secretary. Nothing was done publicly, however, which apparently prompted Wedderburn, a Scottish student at the University of Padua, to reply. 27 In the Quatuor problematum confutatio, Wedderburn brings up Maurolico in the context of accusations that Galileo had taken his design for the telescope from others. Wedderburn first gives the printing press (which, he suggests, was developed independently in Europe and India), and then Rheticus and Maurolico, as examples of independent discovery. 28 The implication of taking an example from Magini himself seems clear: Horky's own patron had argued for such an instance.
Of the early modern references to Maurolico's relationship to Rheticus, Magini seems to have been the most attentive to the actual details of their publication. Fincke may have been following Reinhold (who did not know about Maurolico's table), while Wedderburn was surely more interested in using Magini's claims as ammunition against Horky. In contrast, Magini considered the circumstances of Maurolico's publication, the difference between the two sets of tables, and the availability of Rheticus's canon in his own day. That said, Magini's "evidence" seems far from "extremely clear." However, the entries in Rheticus's and Maurolico's tables are a potential additional clue to help to resolve the question.

Recomputation of tables
Maurolico's treatise contains four single-page tables on folios 65 and 66: tabella sinus recti (sines), tabella foecunda (tangents), tabella benefica (secants), and tabella declinationum et ascensionum (declinations and ascensions). We ignore the latter, except to note that Maurolico uses ε =°23 30' as his value for the obliquity of the ecliptic. All three trigonometric tables give values of their functions for R = 100 000 , , for arguments from 1° to 90°, divided into two columns. 29 Additional columns give the usual differences between successive tabular values for the purpose of interpolation, as well as the differences in sexagesimal notation. We do not reproduce these columns here. The tangent and secant tables are supplemented at the bottom of the page with additional values for arguments 89°15', 89°30', 89°45', 89°55', and 89°59' -the range of arguments where the functions' values change rapidly as they approach infinity. In these two tables, the word infinitum appears opposite the argument 90°.

The sine table
The sine table is recomputed in Table 1. As Glowatzki and Göttsche report, 30 the values are accurate to all decimal places except four (48°, 67°, 73°, 85°), which are all in error by one unit in the last place. Glowatzki and Göttsche say that Maurolico presumably took his sine table from that of Regiomontanus in his Compositio tabularum sinuum rectorum (indeed, they assert that Maurolico's tangent and secant tables were computed from it as well 31 ), which uses a base circle radius R = 10,000,000. 32 The entries for integer arguments in the latter table, given to two more places than Maurolico's table, are remarkably accurate, correct to all seven places except for six entries.

The tangent table
Maurolico's tangents (Table 2) are generally accurate except for small errors in the last place, until near the end of the table. Here Maurolico, along with all sixteenth-century table makers, runs into difficulty. When θ is close to 90° the denominator of tan sin / cos θ θ θ = is close to zero; therefore, any small error in the value for cosθ (due for instance to rounding) will be magnified greatly. This problem would become a major issue later in the century. When Rheticus's magisterial trigonometric tables in the Opus palatinum were published posthumously in 1596, Adrianus Romanus noticed and complained about the poor quality of the tangent and secant values with arguments near 90°, calling one of the entries an "inexcusable error" (perhaps unaware that most other tables of the time were similarly affected). Bartholomew Pitiscus later recomputed the flawed entries. 33 It has been suggested that Maurolico took his tangent values for arguments up to 45° from Regiomontanus's table in his Tabulae directionum, 34 and indeed this seems to be the case, although up to around 60°. Only 25 of the first 59 entries in Maurolico's table are correct to all places (most of the rest are in error by one unit in the last place); all but five of these 59 entries match Regiomontanus's values, including an entry with a large error (probably scribal) at 50°. Beyond 60° Maurolico's values are significantly better than Regiomontanus's; indeed, they are generally slightly more accurate than the rest of the table, even though the numerical instability is greater. As Table 3 indicates, Maurolico's values at the end of the table are substantially more accurate than may be    Table 2. At the end of the table is a note, dated August 13, 1550, explaining that this was a correction of Regiomontanus's table, which, "whether from the carelessness of the author or the negligence of the printers" ("siue authoris Incuria siue Impressorum negligentia) contained multiple errors. 36 These findings suggest that as early as 1550, Maurolico was aware of the issues involved in computing tangents for arguments near 90° and did computational work behind the scenes to evade the problem. Somehow he was able to obtain tangent values more accurate than could be found from any sine table available to him.

The secant table
The entries in Maurolico's table of secants (Table 4), computed for R = 100 000 , , are again determined with great precision. Only 17 of the 89 entries for integer arguments are in error, all by one unit in the last place; and only four of these errors occur in entries with argument greater than 60°, where the function is less stable numerically. The values for integer arguments in Rheticus's 1551 table (see Table 5) are given to two more places. 37 Rheticus's errors do not reach the magnitude of one of the units in Maurolico's table until the argument reaches 85°, 38 so an assertion that Maurolico copied from Rheticus already faces the burden of explaining why fifteen of Maurolico's entries before 85° differ from Rheticus's correct values. But more telling is the fact that the last few entries (especially for arguments 87°, 89°, 89°30ʹ) are an order of magnitude more accurate than Rheticus. Therefore we may reject Fincke's claim that Maurolico copied from Rheticus; indeed, we may credit Maurolico for recognizing the issue of numerical instability and taking extra care when computing these entries.
The explanation for Maurolico's success is partly that he does not compute his secants from a sine table (according to R 2 90 / ) Sin(°−θ ), but rather using the relation sec tan 2 2 2 θ θ = +R . As one may see from the last column of Table 5, Maurolico's  secant table almost perfectly matches computation from his tangent table using this formula. 39 Twenty-five years later, Fincke would use the same strategy within his own tables.

Conclusion
Maurolico's tables, although much smaller than Rheticus's and those of other contemporaries, were among the best of their time. The accuracy of his tangent and secant tables establish Maurolico's independence from Rheticus, but also reveal that his computations were deliberate and sophisticated. Many table makers afterward adopted Maurolico's addition to Regiomontanus's nomenclature for the new functions ("tabula benefica," from "tabula fecunda") alongside Fincke's "secant" and "tangent," while few (apart from Viète) followed Rheticus. The sine, tangent, and secant became the standard three trigonometric functions thereafter. Maurolico's conservative attitude of admiration for the ancient ways, it seems, did not stop him here from helping to forge a new path for trigonometry.   James Steven Byrne is a historian of medieval and early modern science. He is currently completing a book project on the teaching of astronomy in the later medieval arts curriculum.

Notes
1. The versed sine is equal to the radius of the base circle (1 for modern students, a variety of different values historically) minus the sine. 2. "Shadow tables" that tabulate a quantity mathematically equivalent to the cotangent are found during this period, but they did not play roles similar to those played by the primitive trigonometric functions, and were not the origin of what we call the tangent and cotangent functions.