Epitome Astronomiae Copernicanae: Liber VI, part 8
What is an aspect? It is an angle formed by the luminous rays of two planets at the Earth, efficacious for stimulating sublunary nature.
What wins for these angles of rays on the Earth their efficacy upon it? The perfection of proportion between such an angle and the four right angles surrounding one point of the Earth.
In what does the perfection of proportion considered here consist? In two things: 1) that the terms of the proportions be commensurable, that is, that the angle which makes the aspect be a part or aliquot parts of the four right angles, so that such a part measures the sum of the right angles; 2) that the arc which is the measure of this angle be determined or cut off from its circle geometrically, by a line that is either expressible or at least knowable.
What is the reason of the name aspect, and how else are they called? In Greek they are called schematismoi, for the reason that this side which marks off the arc is the side of a schema or regular figure in a circle; but they are called aspects by a poetic or astrological personification, as if all the planets were on the very surface of the sphere of the fixed stars and looked upon one another in this way or that, whereas they are efficacious not on account of arcs of the zodiac but on account of the angles of the rays on the Earth. And in this kind of affection, the situation of the planets, the above-mentioned poetic fiction chiefly prevails among astrologers, who devise that one planet is afflicted, struck, assailed, oppressed by the rays of another, or in turn helped, cherished, relieved, and a thousand other passions.
How many knowable regular figures are there? There are indeed infinitely many by continual doubling of the sides, but the radical ones are only three: the triangle, the square, the pentagon. To the triangle are referred the hexagon, dodecagon and so on, to the square the octagon and so on, to the pentagon the decagon, icosagon and so on, to the pentekaidekagon the triacontagon and so on.
How many aspects are there, then? The old astrology recognises only five: conjunction, when the rays of two planets descending to the Earth are joined in one line; which is as it were the principle of all aspects.
2) Opposition, when the two rays are parts of the same straight line, or when two quarter parts of the circle are intercepted by the two rays, that is one semicircle.
3) Tetragon or quadrate, when one quarter. 4) Trigon or trine, when one third or two sixths. 5) Hexagon or sextile, when one sixth. But meteorological observations have added the semisextile and quincunx from the dodecagon, when namely one or five ounces, that is twelfths of the circle, called signs, are intercepted, so that all the parts of the duodenary division of the circle may furnish their aspects.
To these are added four others from the decagon: the decile, which intercepts one tenth of the circle, the quintile, which intercepts two tenths or one fifth, the tridecile, which intercepts three tenths, the biquintile, which intercepts four tenths or two fifths, with which conjunction and opposition likewise associate themselves, the latter intercepting five tenths, that is a semicircle.
Those physicians also, who for the cure of the sick or for the crises call in astrology, are accustomed, I say, to consider the octagon, in which again occur the conjunction, the octile, semiquadrate or sesquadrate, which intercepts the eighth part of the circle, the quadrate, which intercepts two eighths, the trioctile or sesquiquadrate, which intercepts three eighths, the opposition, which intercepts four.
Which of them are the chief, and what are the ranks of the rest? Conjunction, as the principle of all; opposition, as that which occurs in all three divisions of the circle; the quadrate, as that which occurs in two, the area of the figure being expressible; the sextile, as that whose side is expressible; the semisextile, as that whose side is of the more perfect order among inexpressibles, and which repeated twelve times encloses an expressible plane; and the trine, as that whose side is expressible in power.
Of the second order are the quintile and biquintile, because, although their sides are inexpressible and of the worse condition, they nevertheless share the divine proportion among themselves and the figures themselves excel in congruence into solid figures; to which the quincunx is added, because its figure is fruitful in the congruence of planes, being the companion side of the semisextile. The decile and tridecile, however, already fail in congruence. The lowest are the octile and sesquiquadrate, because they are formed by sides neither expressible nor using the divine proportion, nor in turn relieved by any notable congruence of figure. The other demonstrable figures have now so far receded from the perfection of both kinds that their efficacy vanishes in forming aspects. On these see book IV of my Harmonics.
Compare the chief aspects with the phases of the Moon. The Moon in conjunction with the Sun is new or silent, in the Sun’s sextile is horned, in the Sun’s quadrate is halved, in the Sun’s trine is gibbous, amphikyrtos, in the Sun’s opposition is full. See diagram 66.
How many degrees or signs has each aspect, and how are they numbered?
They are numbered in the great circle which is understood to be drawn through the places of the two planets configured in the sphere of the fixed stars, which is to the ecliptic
Which aspects belong to which planets with one another? With the Sun, Mercury has none besides conjunction, because he cannot run out from it as far as 30°, which is the measure of the least aspect, that is the semisextile. Between the Sun and Venus there can come conjunction, semisextile, decile and octile, besides these none; but between Venus and Mercury, besides these four, also the sextile and quintile. All the other planets in pairs make aspects indifferently.
How do we know when two planets make any one of these aspects? If first, by the latitudes of the planets which they have at the day near the aspect, the quantity of each aspect, already expressed, be reduced to the ecliptic.
Tell the general rules of this reduction. 1) The quadrate aspect of any planet with the Sun or with another which lacks latitude does not need reduction, however great a latitude it itself has. 2) For an aspect less than a quadrant, if the latitudes of the configured planets are evidently different in quarter or even in quantity, a smaller arc of the ecliptic answers, for a greater, a greater. 3) If the latitudes are nearly equal and of the same quarter, then greater arcs of the ecliptic answer to all aspects indifferently.
Teach the method of computing an aspect when the planets have latitude. There are three cases: either one alone has latitude, and then, if the aspect is less than a quadrant, from its antilogarithm, but if greater than a quadrant, from the antilogarithm of the complement to a semicircle, the antilogarithm of the latitude is taken away; the remainder, sought as an antilogarithm, shows the arc of the ecliptic corresponding there to the aspect, here to its complement to a semicircle.
As if the aspect be less than a quadrant, as an aspect of 30°, antilogarithm 14384 Latitude of one 10°, antilogarithm 1531 Remainder 12853, the sought as an antilogarithm, shows the corresponding arc of the ecliptic 28° 26′.
So if the aspect be greater than a quadrant, as an aspect of 144° Complement 36°, antilogarithm 21193. Latitude of one 10°, antilogarithm 1531 Remainder 19662, the sought as an antilogarithm, shows the arc of the ecliptic corresponding to the complement, 34° 46′, which taken from 180° leaves the arc 145° 14′ corresponding to the aspect itself, 144°.
Therefore when the planets are distant in the ecliptic, there by 28° 26′, here by 145° 14′, then in their own peculiar circle they make an aspect, there the semisextile of 30°, here the biquintile of 144°.
Or both have latitudes equal to one another and either of different quarters or of the same: then the aspect being bisected, the reduction proper to the half is sought, as in the first case, if the quarters of the latitudes are different, or to the complement of the half, if the quarter is the same.
Let the aspect be 72°, the latitudes of one 5° north, of the other 5° south. Therefore of the half, 36°, the antilogarithm 20193 is diminished by the antilogarithm of the latitude of 5°, 381. The remainder 19812 as an antilogarithm shows 34° 53′, of which the double, 69° 46′, is the ecliptic distance of the planets making a quintile. But let both latitudes be of the same quarter, then the complement of the half aspect, 36°, is 54°, antilogarithm 53139; from this take away as before 381, the remainder will be 52758, of which the arc is 53° 50′, the complement of the reduction of the half. This will be the ecliptic distance of those making a quintile, if the latitudes are equal and of the same quarter, of 5 degrees each.
Or the latitudes differ in quantity, and the process becomes more laborious; behold the pattern of it.
First if the quarter be the same. Let the latitude of Saturn be 2° 20′ south, of Venus 4° 35′ south, and let them be required to make an aspect of 60°.
Therefore the complement of the greater latitude, 85° 25′, logarithm 320 Difference of the sums 138376, arc 30° 2½′. Half the logarithm 69188. Double 60° 5′. This is the reduction sought.
Again if the quarters be different. Let the latitude of the one be 4° 0′ north, of the other 10° 0′ south, and let them be required to make an aspect of 72°.
Therefore the complement of the greater latitude, 80° 0′, logarithm 1531 Distance of the other from that pole, 94° 0′, logarithm 244 Difference of the sums 108898, 35° 28′. Half the logarithm 54449, 70° 56′. This is the ecliptic distance sought, which if the planets attain, with these latitudes they make a quintile of 72°.
Which of the aspects is of chief observation? Conjunction, as that which in great part falls under the eyes, the planets approaching one another in the same place of the zodiac.
What follows upon conjunctions? Revolutions or restorations of times, various.
By what is any restoration of times defined? By the concurrence of two or more conditions approaching conjunction: for example, if not only the Sun and Moon come together again, but the meeting also falls in the same place of the zodiac as the former, or if not only two meet, but also a third again approaches, or if the Moon not only has rolled out the anomalies of latitude, having returned to the same node, but also falls at once in that return to the node into a conjunction or opposition of the Sun, or into the same place of the zodiac.
How many are the chief kinds of restorations? Two: for either the single planets are referred both to the Sun and to the same place of the zodiac, where the time is sought within which a planet carries round its stations and retrogradations, or the Moon her phases, through the whole zodiac, or the planets are referred to one another and to places of the zodiac.
Recount the restorations of the single planets with the Sun. Saturn and the Sun are restored together to the same place of the zodiac very nearly within 59 years, so that Saturn traverses the zodiac twice, the Sun 59 times. Jupiter within 83 returns of the Sun completes seven of his own. Mars distributes his conjunctions with the Sun at very great intervals. Therefore his restorations are the more accurate the longer they are taken.
The shortest is held to be of 15 solar years, within which he returns eight times; the next of 17 years, in which he completes nine periods; more accurate is that of 32 years, in which 17 revolutions of Mars through the zodiac are made; there follows the period of 47 years, in which are 25 returns of Mars. Still more accurate is that of 79 solar years, having 42 revolutions of Mars, and this above the rest is in esteem among astrologers. Venus returns within 8 solar years five times with sufficient accuracy, and so is joined to the Sun in the superior conjunction in five places, two apart from one another by a space of 72°, at which five places the following conjunctions advance at a very slow pace and by short spaces through eight years each, so that the intervening 72° are scarcely consumed within two centuries and a half and the whole zodiac is frequented by those conjunctions. Mercury on the contrary within 13 solar years completes 54 circuits, wherefore he scatters his conjunctions with the Sun liberally through the whole zodiac, at an interval not greater than 6 or 7 degrees at the mean rate. The Moon, finally, completes twelve returns to the Sun in a somewhat shorter time than the Sun returns to the beginning of his course, and so, that the restoration may become more accurate, a longer time is needed: hence there were born by the observation of the nations the trieteris, embracing 37 lunar months, yet not with the greatest accuracy; the octaeteris, of 99 months, so that the hundredth begins the following octaeteris; the hendecaeteris, of 136 months; and more accurate than all these the one composed of both, the enneakaidekaeteris, of 235 months, and since there still remains a difference of a quarter of one day, by which the new moon fails to return to its former place, Callippus quadrupled the number, establishing a period of 76 years, 940 months.
V. Of the political lunar years.
What is the use of the restorations of the Sun and Moon? Since the Moon by the changing of her phases moves and turns upon herself the eyes of all men, it came about that peoples, seeking signs of times from the Moon, mostly took the beginnings of their years from the beginning of the phases, that is from the new moon, especially those among whom the differences of summer and winter are not very evident, since the return of the Sun to its former places, which defines the year, could not easily be known from elsewhere.
How many kinds of lunar year are there? Two, loose and bound. The loose is of twelve months perpetually, of which one half are a little over thirty days, the rest 29. The first month of such a year in the succession of years runs out into winter, thence into autumn, afterwards into summer, and finally, when the thirtieth year is complete, passes again into spring. Such a year the Turks use, from the discipline of the Arabs. The bound (understand, to the solar year) uses intercalary months and has every fourth or third year thirteen months, so that the first month always wanders about the same season of the solar year, now preceding by some days, now following. The use of these is among the Jews and among ecclesiastical Christians.
How long are these years? Some are called simple and are of 354 or 355 whole days, some embolismic, intercalary, of 384 or 385 whole days, and, as the ingenuity of the art may have been, some also of 353 or 383 days, which days, because after the Jewish manner they begin from the evening, at the rising of the Moon, are for that reason called lunar, and the first Moon is that which is first after the new moon, the rest each by its own number.
In how many ways are the lunar years bound to the solar, and what restorations arise hence? They are bound in two ways, either to the natural quantity of the solar year, or to its political ordinance, such as is the Julian period; and when they are bound to the natural, the restoration or circle is shorter indeed in the enneakaidekaeteris, celebrated by Aratus, longer in that of 76 years, which was the period of Callippus; the longest and most accurate embraces four Callippic periods, that of Hipparchus, for it is of 304 years, within which one day is taken from the reckonings of Callippus.
But when the enneakaidekaeteris or Callippic period of 76 years is accommodated to the ordinance of the Julian year and the observation of the Jewish cycle of perpetual sevens of days, the Julian reckoning of itself contains a cycle of four years, within which, as was said above in the theory of the Sun and in book III, one bissextile day is intercalated, which, four years taken seven times, so that one and the same bissextile and weekday or dominical letter returns, make up the cycle of the Sun, so called, of 28 years. Into this sum, therefore, the number of nineteen years, the cycle of the Moon so called, multiplied, makes the political period of 532 years, named after Dionysius the Abbot its author; after as many years are completed all the bissextiles and weekdays and intercalary months return in the same order, although this cycle is faulty, because it holds the Callippic reckonings, the correction of Hipparchus being neglected, whence it exceeds the reckonings of the Moon by more than a day and a half, but those of the Sun by four days.
What is chiefly to be observed about this great cycle of 532 years? By the observation of this faulty cycle it came about that within 1600 years the place assigned to it in the calendar has strayed from the true equinox by 12 days, and the Moon of the calendar from the Moon of heaven by 5 days. Since many kingdoms and provinces have amended this, so far as was given, it follows that between them and the rest, who hold the ancient reckonings, there frequently arises a difference of Easter, of one, four or 5 weeks, because Easter can be celebrated only on the Sunday after the fifteenth Moon next after the equinox.
What is the golden number? The cycle of the Moon, or the nineteen numbers, assigned by a singular art to the days of the Julian calendar in golden letters; this name they received from the very deed of the first author. But these numbers are indices of the first Moon not always most true, but the usual or artificial one, each in that year of the cycle which it itself indicates in its order.
What was the reason for setting one year before another as first in the decemnovennal cycle? The nearness of the new moon to the equinox of that time at which this ordinance was made, namely before the times of Constantine the Great and before the year of Christ 300, for then the equinox was on the 22nd of Julian March; therefore in the year in which the new moon fell on the evening of the 22nd of March, in that year the 23rd of March was called the first Moon and that year was held for the first, and therefore at the 23rd of March stands the golden number I.
But since 28 decemnovennal cycles constitute one great cycle, by what choice was the first taken, when any might have been first? That decemnovennal cycle was taken first which came nearest with its beginning to the 42nd year of the empire of Augustus: because since Christ in the 15th year of Tiberius was about 30 years old, therefore, these first 15 of Tiberius and the last 15 of the 57 of the empire of Augustus being taken off, one comes to the 42nd year of the empire of Augustus for the birth of Christ, about. But the year which had the new moon on the 22nd of March, nearest to this term, was the 45th Julian or 44th of Augustus. This, therefore, was made the head of the great Dionysian cycle of 532 years, and it happened by a fair chance that it was itself also (or should have been) a bissextile: so that the lunar year began in the next month after the intercalated day, namely from the equinox.
For knowing the cycle of the Moon they bid us add one to the usual years of Christ and to cast out of the sum all the lunar cycles; I ask, whence does it appear that Christ was born in the year 2 of the cycle, the 46th Julian, and what is to be observed about this? That year does not agree with history, but agrees with that artificial cycle according to these two suppositions: first, that Christ was born at the winter solstice, on the 25th of December, which tradition is ancient; second, that Zacharias, the father of John the Baptist, either himself high priest of the Jews or the high priest’s deputy, saw the angel announcing the conception of John the Baptist when he had entered the holy of holies on the day of atonement, which was wont to be the tenth of the seventh month, called Tishri. This hypothesis about Zacharias, who was of the course of Abia, I leave to be examined by theologians skilled in the Jewish affairs of that time. Now if Christ was born at the winter solstice, he was announced nine months before, at the vernal equinox. But the angel affirmed that this was six months after the conception of John; John, therefore, was announced and conceived at the autumnal equinox, about the 25th of September. But he was announced on the 10th of Tishri, or the 10th Moon of the seventh month from the vernal, according to the second hypothesis. But no year of the cycle has the 10th Moon on the 25th of September, or the 1st Moon on the 16th of September, except the first year of the cycle, because the golden number I stands at the 16th of September: therefore in the first year of the cycle John was conceived, in the second year of the cycle Christ was conceived and born, those things being true which I have said. So our era is not historical, but artificial and hypothetical, not too far from the truth. And it is to be noted that today we understand it otherwise than we were taught by its author. He put Christ as conceived at the beginning of the second year of the cycle, beginning from the equinoctial new moon, and born at the following winter solstice, at the end of the 46th Julian year, and therefore they were called of old years not of the nativity but of the incarnation of Christ; but we today suppose Christ born at the preceding winter solstice, while the first year of the cycle is still running, and the 45th Julian year already ending; to us, therefore, they are years from the nativity. This is indeed nearer to historical truth, yet it still strays both from that and at the same time from the reckonings of this cycle.
VI. Of the great and greatest conjunctions.
What chiefly worthy of note occurs concerning conjunctions of the planets with one another indifferently? The conjunctions of the three superior planets, called great and greatest.
What do they call a great conjunction? The conjunction of Saturn and Jupiter, which for the most part lasts so long until Mars also approaches, the third of the superior planets, which is a spectacle to be seen above all, three great stars, shining, without scintillation, in one place of the heaven.
What is the greatest conjunction? When such a conjunction falls about the beginning of the zodiac, which in our zones is the beginning of Aries.
How far apart are pair and pair? One mean conjunction of Saturn and Jupiter, that is of the lines of the mean motion of each, is distant from another by almost 20 years, and in the zodiac by a third of a circle backward, a little less. So it comes about that the fourth conjunction after 60 years surpasses the initial place by about 9 degrees. The greatest, therefore, comes after a little less than 800 years, that is after 794.
How is one period of eight hundred years divided? Into four triplicities or trigons: the fiery, earthy, airy, watery, each of 200 years. For the first 10 conjunctions fall in the signs of the fiery triplicity, Aries, Sagittarius, Leo, as in the years 1584, 1604, 1623 and so on. The following 10 fall in the earthy signs, Taurus, Capricorn, Virgo. The third ten of conjunctions is transferred to the adjacent signs of the airy trigon, Gemini, Aquarius, Libra. The last ten conjunctions are dispersed through the signs of the watery triplicity, Cancer, Pisces, Scorpio. So this field of the zodiac is by this yoke, as it were, of the superior planets ploughed successively in adjacent furrows through the whole, or, more fitly, is frequented on every side by the councils of the planets. See the diagram suited to this.
Distinguish the ages by the greatest conjunctions of the world, adapting known histories. Although the world was perhaps not founded under the very great conjunction of Saturn and Jupiter, for the freedom of these things in the archetype is otherwise, and there is also another kind of things, which here on Earth and in one corner of the world, after the motions have been instituted from the archetype, follows by geometrical necessity, which, being obvious to the senses, we men admire and celebrate as if it were singularly sought in the constitution of the archetype: yet the first great conjunctions that could have been after the origin of the world fell, as chronology shows, about the beginning of the fiery triplicity and the end of the watery. Let the first greatest conjunction, therefore, be at the origin of the world or about the fall of Adam. The second, while Enoch led a divine life on Earth, whose great-grandfather Enos, all the patriarchs surviving, instituted solemn divine worship or the invocation of the divine name, when the Cainites fortified cities and invented arts. The third follows the flood and the new propagation of the human race. The fourth coincides with the exodus of the Israelites from Egypt and the giving of the law. The fifth has the beginnings of the servitude of the Israelites in Media, the kingdom of the ten tribes being extinguished, under Isaias, who delivered most clear prophecies of Christ, when also the Olympiads were instituted and the year of Nabonassar has its beginning and Rome was founded. The sixth has the birth of the promised Messiah from a virgin, in the year of the world 3970 according to some; which is the precise measure of five greatest conjunctions. For 794 years taken five times make 3970. The seventh follows Charlemagne. The eighth, which was marked by a new star, these our times. (Compare vol. II, p. 636.)
VII. Of eclipses or failings of the luminaries, first of the Moon.
What is the notable accident common both to the conjunctions and oppositions of the Sun and Moon and to the illumination of the bodies of the Moon and Earth by the Sun? The privation of light, called in Greek ekleipsis, in Latin deliquium or defect, the chief of those things which the Greeks called pathe, the Latins passions; for so we are accustomed to speak, that the Sun and Moon suffer eclipse, and the poets sing of the labours of the Sun and Moon, and all the other affections of the motions seem to be called passions from their likeness to these. But the Sun is deprived of light only in conjunction with the Moon, the Moon only in opposition to the Sun.
Do the luminaries, then, truly suffer this diminution of light? The Moon is indeed truly stripped of the light which she has from the Sun; but the Sun, although he is covered from our eyes, does not himself lose light, but in his place certain parts of the Earth are stripped of the light of the Sun, not otherwise than the Moon herself before.
Whence does this befall the Moon, that at the time when she ought to be full she is suddenly deprived of light, either wholly or in part of her body? From the interposition of the terrestrial globe between the Sun and the Moon, which we know hence: 1) because it is clear to us that the body of the Earth is opaque and does not transmit the rays of the Sun, but casts a shadow in the opposite direction from the Sun. 2) Because it was said above that the light of the Moon is from the Sun, and by the laws of optics is poured into the Moon by straight lines. 3) Because the Moon never incurs this extraordinary loss of light except in her opposition to the Sun, that is when the Sun, Earth and Moon are in the same straight line and the Earth interposed between the luminaries, so that the Moon stands in the quarter into which the shadow of the Earth extends.
But if this opposition of the Moon is the cause of the defect that occurs, why does the Moon not fail in every opposition? Because not every opposition of the ecliptic places of the Sun and Moon is a true and exact opposition of the bodies themselves also, but frequently the Moon declines from the shadow of the Earth, passing by to the side of it opposite to the Sun.
Where, then, do the defects of the Moon occur, and when? They occur in those places in which the nodes, or head and tail of the dragon, connect the orbit of the Moon with the orbit of the Sun. Since, therefore, the nodes are two, in opposite places of the orbit, and creep with a very slow motion contrary to the order of the signs, as was said, it comes about that the Sun every year arrives at each of them, proceeding in the order of the signs. Wherefore only those full moons which fall when the Sun is near the nodes, in opposite parts of the year, or distant by 6 or 5 months, are in danger as to light; the rest, through the whole year, are out of danger.
What name has the apparent orbit of the centre of the Sun under the fixed stars gained from this? From this event and the passion of the luminaries, chiefly of the Moon, it was called the ecliptic, as if the line of failing or eclipse, of defect; some of the Germans call it the light-breaker, light-loser, light-flier, shadowy, shade-bearing, in the opposition of the Sun, to be sure; while it is meanwhile light-bearing or light-bringing on account of the Sun himself.
Of what figure is the shadow of the Earth? Since both the Sun and the Earth, the former luminous, the latter opaque, are round bodies and the Sun greater than the Earth, it follows by the laws of optics, because the rays of light are straight lines, that the shadow of the Earth is a regular cone or the figure of a round goal, which cone, cut by a plane at right angles to the axis in any point, gets a circle for the section. The same is true much more of the Moon and her shadow.
In diagram 91, if SD be the Sun and YE the Earth, the straight lines SY, DE being drawn touching the Sun and Earth, the cone of shadow YCE is formed. So if KL be the Moon, KRL will be her shadow.
Does the figure of the shadow always remain the same? Not wholly. For the interval of the Sun and Earth being diminished, the shadow becomes shorter, blunter and thinner; that interval being increased, longer, sharper and thicker.
What is the horizontal parallax of the Sun or Moon, and what is the reason of the name? The horizontal parallax is the angle at the Sun or Moon contained by two lines, one touching the surface of the Earth, the other drawn through its centre.
In the diagram VSB is the horizontal parallax of the point S in the Sun, EFB that of the point F of the Moon.
Ptolemy calls it parallaxis, Copernicus from it the commutation of sight, because these lines BS, VS, continued as far as the fixed stars, show two places of the centre of the Sun or Moon, different, under one of which the star is seen as viewed from B, the centre of the Earth, under the other as viewed from V, the surface of the Earth. The sight is therefore imagined to be transferred from B, the centre of the Earth, to its surface V in a moment, in which, were it given, the Sun or Moon would change their apparent place under the fixed stars. For from a low eye, namely in the centre of the Earth B, they would be seen in a higher part of the heaven, from that raised to the surface V they would be seen in a lower place. And so in the Arabic versions parallax is called the diversity of aspect.
How great can the greatest parallax of each star be? From the distances established in book IV, pages 326 and 328, of the Sun 3469, of the Moon 59 semidiameters of the Earth, there follows a parallax, of the Sun, of 59 seconds, or less than one minute in apogee, in perigee as much more than one minute; of the Moon in apogee 58′ 22″, in perigee 63′ 41″.
Teach how to measure the angle at the vertex of the cone of shadow, or its point. If from the angle of the semidiameter of the Sun apparent on the Earth you take away the parallax of the Sun, there will remain half the angle of this point.
As if the semidiameter of the Sun ABS be 15′, BSV or BSC the parallax of the Sun 1′, the half of the point will be 14′, namely SCB or VCB.
Teach also to measure the length of the shadow of the Earth. Multiply the semidiameter of the Earth into the distance of the Sun and Earth, and divide the product by the excess of the semidiameter of the Sun over the semidiameter of the Earth.
As since in book IV, page 326, the proportion of the diameters was a little more than fifteenfold, but the interval of the Sun and Earth 3469 semidiameters of the Earth: take 1′, the semidiameter of the Earth, from 15′ and more, the semidiameter of the Sun; with the remainder, 14′ and more, divide the product of 1 and 3469, that is 3469, and there come out for the length of the shadow of the Earth 247 and more semidiameters of the Earth.
The shadow is therefore more than four times longer than the interval of the Moon and the Earth, which does not become greater than 59 semidiameters, whence it appears that the Moon must at times pass through the shadow.
In the diagram let BC be the length of the shadow, 247 semidiameters of the Earth BE, and BG the height of the Moon in apogee, 59 semidiameters of the Earth BE.
What do astronomers call the semidiameters of the Sun, the Moon or the shadow, and how great are they? So are called the angles at which the semidiameters of the Sun, the Moon, or of the circle of the shadow in the place of the Moon’s passage are seen on the Earth, or which the lines drawn to the ends of those semidiameters, directly opposite, form at the centre of the Earth.
ABD, the semidiameter of the Sun, in apogee is 15′, in perigee 15′ 33″; of the Moon IBL, in apogee likewise 15′ 0″, in perigee 16′ 22″.
Whence is the semidiameter of the shadow had in any place of the Moon’s passage? Let the horizontal parallaxes of the Sun and Moon be thrown into one sum, and from this sum let the apparent semidiameter of the Sun be taken away; there remains the semidiameter of the shadow in that place of the Moon’s passage to which the parallax taken belongs.
As if the semidiameter of the Sun be 15′ 30″, the parallax of the Sun 1′ 1″, of the Moon 62′ 15″, the sum of both 63′ 16″. Therefore the minutes 15′ 30″ being taken away, there remain 47′ 46″ for the semidiameter of the shadow of the Earth GBF in the place of the Moon’s passage FGQ.
What variety occurs in this semidiameter of the shadow? The higher the Moon, the shorter is the semidiameter of the shadow; the lower she is, the longer. In the apogee of the Sun and perigee of the Moon, therefore, it is the longest of all, 49′ 40″, in the perigee of the Sun and apogee of the Moon the shortest of all, 43′ 50″.
In diagrams 92 and 93, AG or AF is the semidiameter of the circle GONMF, by which the shadow is understood to be cut in the place of the Moon’s passage.