Epitome Astronomiae Copernicanae: Liber III, part 10
Thus Tycho Brahe has the Dog Star in the year 1600 at 20° 19′ of Cancer, with latitude 15° 57′ south; its right ascension is 109° 37′, its declination 6° 12′ north. The heaven culminates with 18° 6′ of Cancer. Under the same elevation of the pole the difference of ascensions is found to be 7° 3′, so that the oblique ascension is 102° 34′, rising together with 3° 40′ of Leo, where the angle is 50° 56′. The oblique descension is 116° 40′, setting together with 27° 36′ of Gemini, where the angle is 36° 17′. And since the star is of the second magnitude for Brahe, the depression of the Sun must be fixed at 13°, which, with the angles mentioned, requires arcs of the ecliptic between the rising point and the Sun of 17° 27′ and 23° 14′; the former added, the latter subtracted, give the places of the Sun at the star’s appearing, 21° 7′ of Leo, and at its hiding, 4° 16′ of Gemini. If, however, we follow Ptolemy and number the star among those of the first magnitude, and so take the Sun’s depression as 12°, the star will set when the Sun is two degrees later, and rise heliacally when it is two degrees earlier. Therefore, while the Sun passes between 4° 16′ of Gemini and 21° 7′ of Leo, that is from 10/20 May until 4/14 August, the Dog Star lies hidden under the Sun’s rays. It sets with the Sun on 9/19 June, rises with it on 17/27 July, and culminates with it on 1 July. It sets at the beginning of the day on 16/26 November, and rises at the beginning of the night, that is opposite the Sun, on 13/23 November.
We shall deduce the same things also for the age of Hippocrates the physician, who insists upon the observation of the dog days; not much later than he are Meton and Eudoxus, those whose opinion Aratus followed in the poems in which he described the constellations, of which poems there exist versions into the Latin tongue by Cicero and by Caesar Germanicus. Indeed Eudoxus is constantly on the lips of Pliny and of the writers on husbandry. The fixed stars in the age of Eudoxus were 28° behind where they are today; let Sirius therefore be taken at 10° 36′ of Gemini, and the Dog Star at about 22° 19′ of Gemini. We shall keep the same latitudes, 39° 30′ south and 15° 57′ south. For although the latitude was formerly greater, that was because the obliquity of the ecliptic was likewise greater. Thus, a compensation being made, no error will be committed in right ascension and declination, which are found from the assumed data according to the precepts given above, by means of the table of angles, in this manner.
Arcs of longitude 70° 36′, 82° 19′ from 0° of Aries. These, sought this time not through the degrees of the ecliptic but among the right ascensions, show, as it were, concerning the degrees of the ecliptic these are to be taken by us as arcs of the equator. Under the heading of declinations there correspond to the same the arcs 22° 20′, 23° 20′, which are however to be taken by us as arcs of the circle of latitude. Each is drawn from the ecliptic toward the south. But both latitudes were southern, 39° 30′ and 15° 57′, the former greater than the arc above, the latter less than its arc: from the former, therefore, subtract the arc, and the latter from its arc; there remain 17° 10′ south (sine 29515) and 7° 23′ south (sine 12851), each from the equator; and these are the bases of declination in this case (for in direct use of the table they would be bases of latitude). Thirdly, with the same arcs first set down, there correspond under the heading of angle these angles, 82° 23′ (sine 99117) and 86° 56′ (sine 99857). The sines being multiplied together, as the precept bids, there result the sines of these declinations, 17° 0′ south and For the right ascensions are taken the complements of those angles, 7° 37′ (sine 13254) and 3° 4′ (sine 5350); and they are multiplied into the tangents of the bases of altitude, 30891 and 12958. And the last five figures being cut off, there result the tangents of these prosthaphaeretic arcs, 2° 20′ to be added and 0° 23′ to be subtracted. The former, therefore, is added to the right ascension, because the southern base in Gemini tends forward; the latter is subtracted, because the northern base here tends backward; and they constitute the right ascensions 74° 26′ and 82° 34′.
Let the climate proposed to us be the Dia-Rhodu, the fourth, because it is the middle among the climates numbered by the ancients.
Let the elevation of the pole be 36°, that of the equator 54°. Its tangent is 137638; this, multiplied into the tangents of the declinations, the southern 30573 and the northern 12869, the last five figures being cast off, yields the sines of the arcs of difference, 12° 50′ to be added and 5° 23′ to be subtracted from the right ascension. Therefore the oblique ascensions are these: 87° 16′, rising together with 13° 49′ of Cancer, and 77° 10′, rising together with 4° 37′ of Cancer.
The angles of the ecliptic and the horizon are these: 56° 20′, 52° 8′.
67° 20′, 58° 22′. Let the depression of the Sun be 12° or 13° as before. Hence the arcs between the ecliptic and the horizon are these: 14° 28′ or 17° 12′.
And the upper ones, added to the co-risings, and the lower ones, subtracted from the co-settings, show the places of the Sun for the heliacal rising: 28° 17′ of Cancer, 21° 49′ of Cancer.
For the heliacal setting: 7° 24′ of Taurus, 25° 40′ of Taurus.
On what days of the old Julian year the Sun both today occupies the degrees set forth and formerly occupied them, the following little table shows.
Compare now the testimonies of the ancients. The writings of the ancients have much obscurity, either because astronomy was not yet generally known, or because the variation of the ages had not yet been observed; whence it came about that the astronomers themselves were neither free from errors nor in agreement among themselves. Writers on astronomy, therefore, who are unskilled and follow a leader without certainty, or are in doubt whom to follow, must be read with great attention and discrimination.
Virgil, however, agrees with this reckoning in these verses:
When Taurus with his golden horns opens the year, and the Dog, yielding to the adverse star, sets.
While the Sun, he says, is in Taurus, the Dog sets, yielding to the adverse star. The poet feigns a cause why the Dog sets, namely that it yields to the approaching Sun; and why does it yield? Because the Sun is a star adverse to dogs; he alludes to the dog days, when through the heat, of which the Sun is the cause, dogs are driven to madness, as Pliny has noted. You see that to the heliacal setting of the Dog there is assigned 8° of Taurus and 2 May in the time of Hippocrates, which was about the end of April in the time of Caesar: so the poet, by the word “opens,” alluded also to the name of the month into which this setting would fall, or at least to the month in which the Sun enters Taurus.
Pliny, however, says that the Dog Star rises in Assyria on the sixteenth day before the Kalends of August, and elsewhere has the fifteenth, that is 17 or 18 July; and the little table notes its emersion in the fourth climate on 19 July. Therefore in a more upright sphere, namely in Assyria, it is altogether 18 or 17 July, especially if we make it, with Ptolemy, of the first magnitude, for which reason I have also set down 17 July and, above, 23 May. Pliny, then, looked here to the times of Eudoxus. In another place he involves many things at once, more perplexingly, saying that this happens (he names the rising of the Dog, for the Dog Star) on the fifteenth day before the Kalends of August, the Sun being in the first degree of Leo; whereas our table has 22° of Cancer, and so 8° less than he says. We must prick up our ears to learn what he means by this. For he adds that it happens 23 days after the solstice. Elsewhere he has, 24 days after the solstice, on the eighth day before the Kalends of July, where he names the Dog Star.
Therefore, when he notes 23 or 24 days from the solstice to the emersion of the Dog Star, this agrees with the table for the times of Eudoxus. For the solstice is reckoned at 0° of Cancer, and the table gives 22° of Cancer, almost the complement, an arc which the Sun completes in 23 days. But that he calls this 22° or 23° of Cancer the 1st of Leo has a kinship with the confusion commonly current in that age, which believed that the solstices fall in the eighth parts of the signs. For if the seat of the solstice, which for us is the 1st of Cancer, is transferred by Pliny to the 9th or 8th of Cancer, as is expressed in plain words in book 18, chapter 28, it follows entirely that what our table calls 23° of Cancer is held by Pliny to be the 1st of Leo. Wherefore this remark of Pliny also follows the agreement of the earlier writers.
The heliacal setting of the Dog Pliny describes thus: on the sixth day before the Kalends of May the Dog is hidden in the evening in Attica, that is 26 April; elsewhere, on the third day before the Kalends of May in Assyria, that is 29 April. Elsewhere: nineteen days after the spring equinox, within the space of four days, according to the varying observation of the nations, on the fourth day before the Kalends of May the Dog sets, which the Dog Star must precede. You see that Pliny is speaking of the Greater Dog. Concerning the Dog Star, indeed, Pliny is in vain. For if this precedes in rising, it does not for that reason hold the lead also in setting; rather its setting must follow, for the very reason that it precedes in rising, both limits of its hiding being drawn closer together. But this too is obscure: how can he say here that the fourth day before the Kalends of May is 21 (19) days after the spring equinox? Does he then place the spring equinox here on 9 April, disagreeing with himself, when elsewhere he often puts the equinoxes on 25 March and 24 September and the solstice on 24 June? Or has he twice gone astray here in the perplexities of that age, deceived by the affinity of things? Between the beginnings of the signs and the cardinal points of the year some of the ancients had fixed eight degrees; between the same cardinal points of the seasons and the Kalends of the months Caesar likewise had interposed eight days: so that on the Kalends of the months the Sun held the sixteenth parts of the signs of the old reckoning (the eighth parts according to the true reckoning). Pliny, therefore, assigns to the equinox also the sixteenth day from the true seat of the equinox, descending from the Kalends only as far as he ought to have known that the Kalends had descended from the true seat of the equinox, and as far as Hipparchus made the beginnings of his signs descend from the beginnings of those of the ancients. These manifest errors of Pliny being passed over, let us weigh what bears on the matter, namely that Pliny assigns to the setting of the Dog Sirius the four days following 28 April, whereas the table for the times of Hippocrates puts its hiding on 2 May, which in the time of Pliny would be the last days of April. Here, then, of the four schools which he mentions in his preface, he seems to have followed chiefly the school of Caesar. Again, the same Pliny reports from the teaching of the Egyptians the evening hiding of the Dog on the eighth day before the Ides of May, eight days later, and the morning rising of Procyon, according to the same Egyptians, on the fourth day before the Nones of July (for so it must be read, not June, as what precedes and follows testify). Thus the span of time which he allows in other climates to the hiding of the Dog, namely about 75 days, he reports according to the Egyptians as much shorter, namely 56 days. Can the purity of the Egyptian air, and the diligence of observers in a nation which worshipped Sirius as a god, on whose rising all their hopes depended, avail so much? That is to be considered. For the table puts the rising of the Dog Star with the Sun itself at Rhodes, in the time of Hippocrates, on 1 July, and in the time of Caesar certainly two or three days later; understand it, therefore, not of the emersion but of the rising with the Sun.
What was the reckoning of the dog days among the ancients, what is it today, and the cause of their beginning? Pliny, book II, chapter 17, assigns the rising of the Dog Star to the fifteenth day before the Kalends of August, the day on which he thought the Sun entered Leo, whereas in truth it was in 22° of Cancer. To this day, therefore, by the tradition of the ancients, the beginning of the dog days used to be ascribed. But from the times of Hippocrates to our own this position of the Sun has retreated from 17 July to 2 July. Wherefore the makers of calendars, disregarding the passage of the fixed stars into the following signs, have fixed this seat for the dog days, some at the most ancient date, 17 July; others, adapting it to the seventh climate, have transposed it to 19 July and retain it even today, just as if in the Julian year either the Sun or the fixed stars returned; others, considering the difference of the natural year from the Julian civil year, have each in his own age weighed the measure of the anticipation of the natural year in the Julian, and, despising the fixed stars from which the name arose, have made the dog days ascend with the Sun itself in the Julian year. Wherefore in the calendars of today, according to the custom which each follows, the dog days begin either on 19, or 17, or 16 July, or 7 or 6 July. For after the Gregorian correction, ten days having been removed, that most ancient seat on 17 or 16 July rose in a single moment to 7 or 6 July, the dog days being left on 17 and 16 of the new calendar; which was done not badly with regard to the reckoning of the natural year, since these days, 17 and 16 July, have here been drawn back nearer to the 22° of Cancer which the Sun formerly occupied. But others, thinking every most recent observation in the calendars weighty and almost sacrosanct, and not attending to its first origin, lest by the removal of ten days violence should be done to the heavens and to nature, have transposed the dog days from 16, 17 or 19 July, just as these days have slipped through so many centuries, to the corresponding days of the Gregorian year, 26, 27 or 29 July. Others in turn, having carried them over by the removal of ten days to 6 and 7 of the Julian July and having used them so for some time, have removed them to 6 and 7 of the Gregorian July, or 26 and 27 June of the Julian. So much, then, concerning the custom received in the calendars and the civil year.
But if we look to the nature of the sidereal and natural year, two beginnings of the dog days arise for us today: the one has the same heat as the Hippocratic, the other the same star and name. For to 22° of Cancer, the place where the Sun formerly discovered the Dog Star, it returns today on 4/14 July, or, if the ancient rising of the star were reckoned in our climate, on 6/16 July. But in our climate and our age the Dog Star is found to emerge, as was said above, on 4/14 August. Now it rests with those who consult nature to decide to which they will assign the force of the dog days, to the height of the Sun or to the star of the Dog. For if even today, though the two are sundered, they still hold that the forces are tempered by an equal interval of days, the first seat will be kept on 17 or 19 July of the Julian civil year, 27 or 29 of the Gregorian.
How great is the number of the dog days, and what is the reason of that number? Although here too the calendar-makers differ no less, some counting 30, others 34, others 40, others 42, yet anciently 45 days were fixed, not now by observation of the heavens but of the weather, since for as many days the Etesian winds blow after the rising of the Dog, as the ancients observed. Pliny places the forerunners of the Etesian winds on the eighth day before the Kalends of August, 7 days after the rising of the Dog Star, at the very rising of the Dog Sirius, and their end in Assyria on the fifth day before the Kalends of September, in Egypt on the sixteenth day before the Kalends of October, thus 34 or 54 days, of which the mean is 44. Yet the whole time between the stars of the Dog and of Arcturus seems to be attributed to the Dog, because the nature of the year during that time remains almost in its own state, until the approaching morning rising of Arcturus brings (in the judgment of observers) a contrary and evident tempest: for the rising of Arcturus too is called by Pliny a violent star.
Examine also the rising and setting of Arcturus, but for the age of Hesiod, who wrote of it. From the table above it appears that for the 2400 years from Hesiod to us a motion of the fixed stars of about 34 degrees is due, which, subtracted from the present place at 18° 40′ of Libra, leave 14° 40′ of Virgo. By the former method I seek its right ascension and declination thus.
Let the same elevation of the equator, 54°, be taken for Rhodes, and let its tangent, 137638, divide the tangent it sets at 14° 32′ of Scorpio.
The angles at these points are 74° 58′ and 35° 0′; let their sines, 96578 and 57358, divide the sine, the former to be added to the co-rising point, the latter to be taken from the co-setting point, so that the places of the Sun are shown as 19° 50′ of Virgo. Hence the table of the appearances of Arcturus for the age of Hesiod.
Compare now the verses of Hesiod.
He expressly states the evening rising of the star from the horizon, or acronychal, and adds the sixtieth day after the winter solstice. You see, however, that it was then accustomed to rise in the evening when the Sun was in 5° of Pisces, that is 65° after the winter solstice. You have, then, the testimony of Hesiod that the solstice falls after the beginning of Capricorn by 5 days and more, if we suppose a smaller motion of the fixed stars between the ages of Hipparchus and Hesiod, as Hipparchus and Ptolemy held; and it can be believed that this too was among those documents from which Eudoxus or Hipparchus reasoned that the solstices fall in the eighth parts of the signs; but of this matter more in book VII.
Pliny places the rising of the star 11 days before the equinox, which agrees nearly with the times of Hesiod, for you see that the Sun at the emersion of Arcturus is placed more than 10° before the equinox. He mentions its setting three times: first on the eighth day before the Ides of June, which is true today of the morning setting, which you see falls on 6 June of the Julian calendar; again the same on the eighth day before the Ides of August, which is plainly absurd; thirdly the evening setting, on the day before the Kalends or the fifth day before the Nones of November, which alone agrees. For Pliny is distant from Hesiod by a third of the interval of time, and from us by two thirds.
For us also Arcturus is hidden on 25 November, for Hesiod on 25 October; therefore by analogy for Pliny on 5 November.
Of the antoeci, perioeci and antipodes.
Whom do the geographers call antoeci, whom perioeci, and whom antipodes? Antoeci and perioeci are so called as though “dwelling opposite” and “dwelling around.” The antoeci dwell under the same terrestrial meridian, but in different climates, equally distant on either side from the equator; as L and O (figure 44). The perioeci, on the contrary, dwell in the same climate, but at opposite intersections of it with one and the same meridian, as L and P. The antipodes dwell both in different climates, equally removed from the equator, and at opposite intersections of those with one meridian, that is, diametrically opposite one another, as L and A. Some are accustomed to call those whom I call antoeci by a name compared with that of the antipodes, because, as the feet of the latter, so the shoulders of the former are opposed and turned toward each other; and sometimes they are accustomed to name the perioeci and the antipodes indiscriminately antichthones.
By what properties of their appearances are they distinguished from one another? For the antoeci the differences of the seasons of the year are contrary; for the perioeci the differences of day and night are contrary; for the antipodes both. For the antoeci have noon at the same moment and midnight at the same moment, but summer and long days belong to the one when winter and long nights belong to the others, and conversely. Except the torrid zone, where with days and with summer it is as I have said; there they can have winter at the same time, but not equally constituted, nor of the same, but of a contrary character. Likewise except something in the frigid zones, when there is continuous day, which because of the Sun’s proper motion falls out longer for the one than for the others. For the perioeci days and nights occur at the same time and equally long; yet they have the middle of the day when the others have the middle of the night, whether true night or something analogous to midnight. For the antipodes all things occur contrariwise: summer and long days, or no nights, when the others have winter and short days or none; night there when it is day here, and day when it is night here. Moreover, at the same moment at which the Sun rises for us, it sets for them, and when it sets for us, it rises for them.
Add to this, from what was said above, that for the antoeci and the antipodes, being inhabitants of the southern hemisphere, the circumstances of the rising and setting of the stars, and of the ascensions and descensions of the signs of the ecliptic, are contrary. For the stars which always appear to us are never seen by our antoeci and antipodes, and those which we never see they always see. The remaining stars interchange their risings and settings, and are common to us and to them, but in a different part of the diurnal revolution: for with the same degrees of the equator and ecliptic with which the several stars rise for us, they are hidden for them, and conversely. Evidently because the signs of the ecliptic also, which for us are in the ascending semicircle, constitute for them the descending semicircle.
What is to be observed concerning the antoeci, perioeci and antipodes of those who dwell in the right and the parallel sphere? Of those who dwell in the right sphere there are no antoeci, and the antipodes are among the perioeci. So also those who dwell in either parallel sphere, that is, those who live under either pole, are both antoeci and antipodes to one another, and have no perioeci.
Is there also any diversity of shadows among the antoeci, perioeci and antipodes? None at all. For each of the pairs are either amphiscian, like the antoeci, perioeci and antipodes of the torrid zone, or periscian, like those of the frigid zones, or heteroscian, like those of the temperate zones. In comparison with the human body, however, the same happens to the antoeci and the antipodes with regard to shadows as was said above happens to them with regard to the risings and settings of the Sun. For the shadow seems to go to the left for the one and to the right for the other.
Of the longitudes and distances of places.
Since you have everywhere compared the right ascension of the stars with the longitude of places on the Earth, and the declination with their latitude, say then to what end the knowledge of the longitude and latitude of places on the Earth conduces. 1) It is a part of astronomy concerning the differences of longitudes of places on the Earth, or the difference of meridians in hours, so that the celestial phenomena can be adapted to different meridians. 2) Without the longitudes and latitudes of places the distances of places cannot be investigated by calculation.
How much time is one degree of difference of longitude on the Earth worth? Since the Sun, returned to the meridian of the same place from which it had departed, fixes the limit of the natural day and night, of 24 hours, therefore 15 degrees of the terrestrial equator are worth one hour, and one degree of time or arc is worth four minutes of an hour, and so on.
How is the difference of longitude of places on the Earth investigated? From the elevations of the pole and the distance of the places. For if the itinerary distance of the direct road is known in German miles, it is converted into degrees of one great circle, drawn through both places, taking 15 miles for one degree; or if one place can be seen from the other, the arc of the great circle between the places is investigated by the method that was given in the first book. Then is formed the triangle SIP (figure 45) between the two places S, I and the pole of the Earth P, with its sides known; and the procedure for finding the angle at the pole, or the difference of longitude, is plainly the same as it was above when the difference of ascension was sought, likewise from the distance; and so there is need only of examples.
Let the places be of the same latitude.
Of Linz the elevation of the pole, or latitude of the place, is 48° 16′, the complement 41° 44′ Both northern. Between Linz and Graz there are counted at most 30 miles, that is 2 degrees in a great circle.
remainder 90799, half 45400 (if the sum exceeded a quadrant, you would add the excess of the sine).
Subtract the sine of the complement of the distance, 2°, namely 99939, from the first sine 99978; the remainder is 39, to be continued with 5 ciphers (if the distance exceeded a quadrant, you would add the excess of the sine).
3900000: 45400 = 86; the quotient is a versed sine, therefore the sine is 99914, the complement 87° 37′ 30″.
This is the arc of the terrestrial equator sought, or the angle at the pole, the difference of longitude between Graz and Linz. In time it is worth minutes of one hour. By so much, therefore, more time is counted at Graz at any moment than at Linz.
Let the places be of different latitudes. As if some Christian from the city of Pemba in the kingdom of Congo, situated on the southern and western coast of Africa, should undertake a pilgrimage of devotion to Jerusalem, and say that he has travelled 708 German miles of direct road, what will be the difference of longitudes, since we know that the latitude of Jerusalem is 31° 40′ north, and that of Pemba about 8° 10′ south? Therefore 708 miles are an arc of 47° 12′ of a great circle.
(If the sum were less than a quadrant, you would subtract the complement of the sine as before.) The aggregate is 168497.
The sine of the complement of the distance, 42° 48′, is 67940; add the sine 91706 (if here there were an excess, you would subtract, contrary to what was done before). The sum is 159646: 84249 = 189493, which is the versed sine of the arc 153° 30′. The complement of this, 26° 30′, is the difference of longitudes sought. Therefore Pemba would be 26° 30′ west of Jerusalem, that is 1 hour 46 minutes.
How is this difference of time to be used? When it is certain that some celestial phenomenon occurs at the same moment in all places of the Earth, as when the Moon begins to enter the shadow, then if the time of the phenomenon is had at a certain place, those who are more westerly than that place always count less from noon, those who are more easterly count more. Therefore the difference of longitudes in hours is to be subtracted for the more westerly, and added for the more easterly.
Name some marvellous phenomenon resulting from this cause. 1. The Portuguese, of whom mention was made in book I, folio 128, after a voyage of three years spent in their circumnavigation of the globe, returning home, counted Wednesday when Portugal was keeping Thursday: for, following the Sun, they carried the meridian of their ship, from which they counted the beginning of the natural day, once around, while the meridian of Portugal had remained in its place. The contrary befalls those who set out toward the east and return home from the west, for meeting the Sun they count in the end one day more than those to whom the return is made. 2. This question exercises the Jews: if two Jews, one going through the west and the other through the east, penetrate into China and meet each other, whose sabbath is valid? For they will bring different sabbaths (differing by one day), and he who comes by the east will be ahead. (Compare IV. 39.) Indeed even Christians, rigid observers of their festivals, will be vexed by the same perplexity; since all festivals begin, for the Jews at sunset in any place, so for the Christians at sunrise.
By what other methods is the difference of longitudes sought? The astronomical means are only two, one by the observation of eclipses of the Moon, already explained, the other by the visible place of the Moon when it stands at the ninetieth degree of the ecliptic, of which it is not the place to say more here; each serves for places far distant. The natural philosophers attempt one method by means of a round magnet, the image of the Earth, whose principles are as yet uncertain; for it is required that, as the axis of the natural magnet directs itself of its own accord into the plane of the meridian, and as the pole of the magnet dips of its own accord deeper or higher according to the nearness of the place to the pole of the Earth, so also there should be in the middle circle of the magnet a certain part which turns of its own accord toward its native place. Whatever the truth of the matter, it is likely that the observation can be only a coarse one, and the less useful for that.
Book Three of the Epitome of Astronomy. Part Five
There is also one chorographic method for places not very far apart, namely those one of which can be seen from the other, so that the elevation of the pole of each place is known, the meridian line of the one, and the remaining azimuth, counted from its meridian, which angles they call angles of position. There is given thus in the former triangle SIP the angle PSI with the two sides PS, PI comprehending the angle SPI sought: which method sailors imitate to some extent by the use of the compass card, if they sail continually in the quarter of the same wind and observe the elevation of the pole both at the place from which they set out and where they put in. But because they do not sail on a great circle (keeping the same quarter) but on a spiral of the globe, let us therefore dismiss this method as merely geographical.
Show how the distance of places may be made known otherwise than by travelling. This can be done, among other ways, also by the aid of astronomy, from the latitudes of two places and the difference of longitudes, in the same triangle SPI as before; the procedure is the same in the first part, but contrary in the second, much as when above the declination was sought from the distance of the star and of the pole from the vertex and from the angle of azimuth.
The procedure is this.
Let the latitudes be of the same name. For example, let a ship setting out from Coimbra in Portugal be bound for America, to put in at the mouths of the river Panuco in the Gulf of Mexico; and let it be known from astronomical observation that the difference of meridians is 6 hours 16 minutes. Let the latitude of Coimbra be 40°, of the mouths of the Panuco 23°, both northern.
Therefore the complement of the smaller side is 67°, of the greater 50°, the sum 117°, the excess 27°. 23° + 50° = 73°, whose sine 95630 add to the sine of 27° (45399); you would subtract the sine of the complement. The aggregate is 141029, half 70515.
The difference of meridians resolved into degrees makes 94°. Multiply the versed sine of this angle, 106976, by 70515 (the last 5 figures being cast off); the product is 75434; subtracted from the first sine 95630, there remains 20196, the sine of an arc of 11° 37′; its complement 78° 23′ is the distance of the places on a great circle. Taking 15 miles to the degree, there are made up 1175 and three quarters miles.
Let the latitudes be different. As if it be asked how many miles there are from the Cape of Good Hope at the extreme southern corner of Africa, whose latitude is 35° south, to Diu in Cambay, a fortress of the Portuguese at the mouths of the river Indus, whose latitude is 21° north. Let it be established from observation of eclipses that the difference of meridians is 4 hours 4 minutes, that is 61 degrees of the angle at the pole, or of the terrestrial equator.
The complement of the smaller latitude is 69°, the smaller latitude 21°. The complement of the greater latitude is 55°.
The excess is 34°, the sine 55919 (you would subtract the sine of the complement). The aggregate is 152949. Half is 76475.
Let the versed sine of the complement of the angle 61° to a semicircle (119°), namely 148481, be multiplied by 76475, the last 5 figures being cast off. The product, the greater, is 113550 The first sine is 97030 The difference 16520 is the sine of an arc of 9° 30′. An excess above 90°, because the product was greater, therefore 99° 30′. Because the latitudes are different, this arc is not the distance sought, but its complement to a semicircle, 80° 30′. It is worth 1207 miles.