Tome II
by Monsieur De La Loubere
Envoy Extraordinary from the French King, to the King of Siam, in the years 1687 and 1688. Wherein a full and curious Account is given of the Chinese Way of Arithmetick, and Mathematick Learning. In Two Tomes, Illustrated with Sculptures. Done out of French, by A.P. Gen. R.S.S.
1693
Tome II, p. 186-199
Rules of the Siamese Astronomy, for calculating the Motions of the Sun and Moon, translated from the Siamese, and since examined and explained by M. Cassini, a Member of the Royal Academy of Sciences.
Monsieur de la Loubere, the King's Ambassador extraordinary at Siam brought back a Siamese Manuscript, which comprehends the Rules for calculating the motions of the Sun and Moon, according to the method of that Country, the Translation thereof he likewise brought from Siam, and communicated unto me.
This method is extraordinary. They make no use of Tables; but only the Addition, Subtraction, Multiplication, and Division of certain numbers, of which we do not presently discern the Ground, nor to what these numbers refer.
Under these numbers are conceal'd divers Periods of Solar Years, of Lunar Months, and other Revolutions, and the Relation of the one with the other. Under these numbers are likewise conceal'd several sorts of Epoches which are not distinguished, as the Civil Epoche, the Epoche of the Lunar Months, that of the Equinoxes, Apogaea, and Solar Cycle. The numbers in which the difference between these Epoches consists, are not ordinarily at the head of the Operations to which they serve, as they ought to be according to the Natural Order: they are often mixed with certain numbers, and the Sums or differences are multiplied or divided by others, for they are not always simple numbers, but frequently they are Fractions, sometimes Simple, sometimes Compound, without being ranged after the manner of Fractions, the Numerator being sometimes in one Article, and the Denominator in another; as if they had had a contrived design to conceal the Nature and Use of these numbers. In the Calculation of the Sun, they intermix some things which appertain only to the Moon, and others which are not necessary, either to the one or to the other, without making any distinction. They confound together the Solar and the Lunisolar Years, the Months of the Moon, and the Months of the Sun, the Civil and the Astronomical Months, the Days Natural and the Days Artificial. The Zodiack is divided sometimes into twelve Signs, according to the number of the Month of the Year, sometimes into 27 parts, according to the number of the Days that the Moon runs through the Zodiack, and sometimes in 30 parts, according to the number of the Days, that the Moon returns to the Sun. In the Division of the Day there is no discourse of Hours; but therein is found the 11th the 703d and the 800th parts of the Day, which result from the Arithmetical Operations which are prescribed.
This Method is ingenious; and being illustrated, rectified, and purged from Superfluities, it will be of some use, being practicable without books, by the means of divers Cycles, and of the difference of their Epoches. Wherefore it is that I have endeavoured to decypher it, what difficult soever I found at first, not only by reason of the confusion which everywhere appeared, and of the Names which are wanting in the supposed numbers; but likewise by reason of the extraordinary names, which are given to what results from the Operations, of which there are more than Twenty which have not been interpreted by the Translator, and of which I could never have found the Signification, if I had not first discover'd the method; which has likewise evinced to me, that the Interpretation, which the Translator has made of three or four other names, is not very exact.
In this research I have first distinguished, and separated from the other numbers, those which belong to the Epoches, having observed that these numbers, are those which were given to add or to subtract, either simply, or by dividing or multiplying them by certain other numbers.
Secondly, I have considered the Analogies which result from the Multiplications and Divisions of the other numbers, separated from the Epoches; and it is in the terms of these Analogies, that I have found the Periods of the Years, of the Months, and of the Days, and the differences of the one from the other, which the experience of things Astronomical, and the occasion of divers operations which I have made, has been me to understand.
I thought that the Missionaries, to whom Astronomy gives admittance amongst the great and learned throughout the East, might reap some advantage from this work, for the Understanding, and for the Explication of the Oriental Astronomy, which might easily be rectified and adapted to ours, with a little altering the Method, by correcting the numbers which it uses.
I thought also that it would not be useless to reduce the Astronomy of Europe to this form, to be able to supply the want of the Tables which greatly abridge the work. This method would be much more easie to practice in the form of the Julian and Gregorian year of which we make use, than in the form of the Lunisolar year, which the Orientals observe: for their principal difficulty consists in reducing the Lunisolar years and the Civil Lunary months to the years and months of the Sun, which the form of our Kalender immediately gives us; and what has given me the most trouble, has been to find out the method which they use to reduce them, in which the several sorts of Years, Months and Days, which are supposed and sought, are not distinguished. Wherefore the reason of the Explication which I give, and of the Determination of the Genus to the Species which I make in the beginning, will not presently be understood; but in the sequel it will be comprehended by the Connexion of things, and by what necessarily results therefrom.
The jurisprudence of the Hindus and Arabs being the field which I have chosen for my peculiar toil, you cannot expect that I should greatly enlarge your collection of historical knowledge; but I may be able to offer you some occasional tribute; and I cannot help mentioning a discovery which accident threw in my way, though my proofs must be reserved for an essay which I have destined for the fourth volume of your Transactions. To fix the situation of that Palibothra (for there may have been several of the name) which was visited and described by Megasthenes, had always appeared a very difficult problem, for though it could not have been Prayaga, where no ancient metropolis ever stood, nor Canyacubja, which has no epithet at all resembling the word used by the Greeks; nor Gaur, otherwise called Lacshmanavati, which all know to be a town comparatively modern, yet we could not confidently decide that it was Pataliputra, though names and most circumstances nearly correspond, because that renowned capital extended from the confluence of the Sone and the Ganges to the site of Patna, while Palibothra stood at the junction of the Ganges and Erannoboas, which the accurate M. D'Ancille had pronounced to be the Yamuna; but this only difficulty was removed, when I found in a classical Sanscrit [Sanskrit] book, near 2000 years old, that Hiranyabahu, or golden armed, which the Greeks changed into Erannoboas, or the river with a lovely murmur, was in fact another name for the Sona itself; though Megasthenes, from ignorance or inattention, has named them separately. This discovery led to another of greater moment, for Chandragupta, who, from a military adventurer, became like Sandracottus the sovereign of Upper Hindustan, actually fixed the seat of his empire at Pataliputra, where he received ambassadors from foreign princes; and was no other than that very Sandracottus who concluded a treaty with Seleucus Nicator; so that we have solved another problem, to which we before alluded, and may in round numbers consider the twelve and three hundredth years before Christ, as two certain epochs between Rama, who conquered Silan a few centuries after the flood, and Vicramaditya, who died at Ujjayini fifty-seven years before the beginning of our era.
-- Discourse X. Delivered February 28, 1793, P. 192, Excerpt from "Discourses Delivered Before the Asiatic Society: And Miscellaneous Papers, on The Religion, Poetry, Literature, Etc. of the Nations of India", by Sir William Jones
Concerning the Astronomical Epocha of this Method.
I have endeavoured to discover what is the Epoche, from whence they here begin to compute the Motions of the Sun and Moon; and to what year, what month, and what day of our Kalender it refers: for it is not treated of in this extract, which supposes it either known, or explained perhaps in the preceding Chapters from whence this extract has been taken, seeing that without the knowledge of the Epocha, it is absolutely impossible to practice this Method.
I have found that this Epoche is Astronomical, and that it is different from the Civil, which I have understood, because it is here prescribed to begin to compute the Months of the Year, current with the fifth Month in the Leap Year, which consists of 13 Months, and with the sixth Month in the common Year, which consists of 12 Months. For this would not be intelligible, if they supposed not two different Epoches of Years, the one whereof, which must be the Astronomical, begins sometimes in the fifth, and sometimes in the sixth Month of the other, which is the Civil. That which likewise evinc'd to me that the Astronomical Epoche, is different from the Civil Epocha, not only in the Months, but also in the Years, is the Operation which is here made to find the Year of ones Nativity, by subtracting his Age from the number of the Years elaps'd since the Epocha; for this Operation would be useless, if they demand only the Year of the Birth after the Civil Epocha, which is immediately known, and which is compared to the Year current, to know the Age of a Person.
This being supposed, I have first searched out the Age to which this Astronomical Epocha may refer; and having found in the Calculation of the Sun performed by this method, that two Signs and twenty Degrees which are therein employed, can only denote the place of the Zodiack, where was found the Apogaeum of the Sun in the Epocha, which Apogaeum must be in the twentieth Degree of Gemini; I judged that this Epocha must be about the seventh Age, where the Apogaeum of the Sun is found in the twentieth Degree of Gemini according to most Astronomical Tables.
Secondly, having found that the number 621, which is intermixed in the Calculation of the Sun, can only be the number of the days comprized, between the Astronomical Epocha, and the return of the Moon's Apogaeum, to the beginning of the Zodiack; and that the number 3232, which is afterwards employed therein, can be only the number of the Days, during which, this Apogaeum makes a Revolution; I have confirmed that the Apogaeum of the Moon which in 621 Days makes two Signs and nine Degrees, was in this Epocha, in the 21 Degrees of Capricorn: And because that the Moon's Apogaeum by the Revolution it makes in eight Years three quarters, returns to the same degree of the Zodiack twelve times in one Age; I have distinguished the Years of the Age, in which the Moon's Apogaeum is found in this Degree, and I have excluded the other Year.
Thirdly, having found by the method here used for Calculating the place of the Sun, that this Astronomical Epocha is very near the vernal AEquinox, which in the seventh Age fell on the 10th or 21st of March; Amongst these select Years I have found one in which the Moon's Apogaeum, arrived at this Degree of Capricorn, about the 21st of March, which is found but once in 62 Years, wanting some Degrees; and I have found that in the 638th Year of Jesus Christ, the Apogaeum of the Moon was at the 21st Degree of Capricorn the 21st of March.
Fourthly, I have remarked that this Astronomical Epocha must have begun at a new Moon; because the Lunar Months are reduced into Days, to find the number of the Days from the Epocha, and the value of the whole Months being deducted from the Sum of the Days, the test serves to find the Moon's distance from the Sun.
In the 638th Year of Jesus Christ, the AEquinoxial new Moon happened teh 21st of March at three a Clock in the Morning at Siam, when the Sun by its middle Motion ran through the first degree of Aries, the Sun's Apogaeum being in the 20th Degree of Gemini, and the Moon's in the 21st Degree of Capricorn. This Day was likewise remarkable for a great Eclipse of the Sun, which happened the same day, but 14. Hours after the mean Conjunction.
Fifthy, By the manner of find the day of the week, which is here observed, it appears that the day of the Epocha, was a Saturday, and th3 21st of March, in the Year 638 was also a Saturday. This likewise confirms the certainty of this Epocha, and demonstrates the Knowledge and Judgment of those that have established it, who contented not themselves with a Civil Epocha, as other astronomers have done: but who have chosen an Astronomical one, which was the Natural Principle of several Revolutions, which could not begin again, till after several Ages. This Epocha is 5 Years and 278 Days distant from the Persian Epocha of Jesdegerdes, the first year of which began on the 16th of June, in the Year of Jesus Christ, 632. Yet these Indian Rules are not taken from the Persian Tables related by Crosotoca; for these Tables do make the Sun's Apogaeum two degrees more backward, and the Moon's Apogaeum above six degrees forwarder; which agrees not so exactly with our modern Tables. The Persian Tables do also make the Sun's AEquation 12 Minutes less, and that of the Moon 4 Minutes greater; which agrees better with the Moderns.
These Indian Rules are not drawn neither from the Tables of Ptolemy, where the Sun's Apogaeum is fixed to the 5th degree and a half of Gemini; nor from the other Tables since made, which have all this moveable Apogaeum. It seems therefore that they have been invented by the Indians; or that perhaps they have been taken from the Chinese Astronomy, as may be conjectured from this, that in this extract the Numbers are written from the top downwards, after the manner of the Chineses: but it may be that this way of writing the numbers might be common to these two Nations.
Having found the Astronomical Epocha of this method, and the Relation is has with the Julian years; we may rectifie teh Epocha's of the motions of the Sun and Moon by the modern Tables, by adding about a Minute a Year to the Sun's apogaeum, and by correcting the other Periods. Thus there will be no difficulty, to reduce the Years and Months since the Epocha into days; and if the Equations are likewise corrected conformably to the modern Tables, we shall by the same Method, find the place of the Sun and Moon with a great deal more exactness. We will give this Correction, with the Supplement of what is wanting in these Rules, after that we have explained them.
Rules to find the place of the Sun and Moon at the time of any Person's Birth / Explication.
I. / I.
1st. Set down the AEra / 1st. The Aera in this place is the number of the years since the Astronomical Epocha from whence is take the motion of the Planets to the current year; which will appear in the sequel.
2nd. Subtract the Age of the Person from the AEra, you will have the Age of the Birth. / 2nd, The Age of the Person, is the number of the Years from his Birth to the Year current, which being deducted from the AEra, there remains the Age or time of the Birth, that is to say, the Year from the Astronomical Epoche in which the Nativity happened.
3rd. Multiply it by 12. / 3rd. By multiplying the years by 12 they are reduced into Months. These Months will be solar, each consisting of 30 days, 10 hours and a half, a little more or less, according to the several Hypotheses, if the years are solar; or near upon if they are lunisolar, and in so great number, that the excess of the one recompences the defect of the others.
4th. Add hereunto the number of the Months of the year current1 and for this purpose if the year current is Attikamaat, that is to say, if it has 13 Lunar months, you shall begin to compute with the 5th month; but if it is not Attikamaat, you shall begin to compute with the 6th month. / The Form of the Year here mentioned, is lunisolar, seeing there are some common of 12 lunar months, and abundant or Embolismal, called Attikmaas, of 13 lunar months. For that they begin to compute the months, not with the first month of the year, but with the fifth, if it is Leap year, and with the sixth if it is not: I have inferred that there are two Epocha's, and two forms of different Years, the one Astronomical and the other Civil: that the first Month of the Astronomical Year begins in the fifth Month of the Civil Leap year, which would be the sixth Month without the intercalation of the Leap month, which is not reckoned amongst the 12 Months, and which is supposed to be inserted before; and that in the Other Years, all the Months of which are successively computed without Intercalation, the first Month of the Astronomical Year, is computed only from the sixth Month of the Civil Year.
But as it is not expressly determined here, whether one ought to begin to compute an entire month at the beginning or end of the 5th or 6th month, it may be that for the first month of the Astronomical Year they take, that which ends at the beginning of the months whereof it is discourses in this Article. In this case, the Interval between the beginning of the Civil Year, and the beginning of the Astronomical Year, would be only of 3 or 4 entire months: whereas if an entire month is reckoned only at the end of the 5th or 6th month, and that the first month which is reckoned, according to this Rule, be the first of the Astronomical Year; the interval between the beginnings of these two sorts of years, will be 4 or 5 whole months. We shall see in the sequel, that the Indians have diverse sorts of Astronomical Years, the beginnings of which are different, and are not much distant from the Vernal AEquinox; whereas the Civil Year must begin before the Winter Solstice, sometimes in the month of November, sometimes in the month of December of the Gregorian Year.
They add the number of the months of the current year, which are lunar months, to those that they have found by the third Article, which are solar months; and they suppose that the sum, as heterogeneous as it is, should be equal to the number of the solar months elapsed from the Astronomical Epochs. They neglect the different that there may be, which in a year cannot amount to an entire month: but they might be deceived a month in the succession of the years, if they took not good heed to the Intercalations of the months, after which the number of the months which are computed in the Civil Year, is lesser than that which they would reckon without the precedent Intercalations.
5th. Multiply by 7 the number found Art. 4.
6th. Divide the sum by 228.
7th. Joyn the quotient of the division to the number found Art. 4. This will give you the Maasaken (that is to say, the number of the months) which you shall keep.
-- / 5th, 6th, 7th. They here seek the number of the lunar months from the Astronomical Epocha, discoursed of in the 1st Article, to the beginning of the current month: which is performed by reducing the solar months, which are supposed to have been found above, into lunar months, by the means of the difference, which is between the one and the other. In the operations which are made, it is supposed that as 228 is to 7, so the number of the solar months given, is to the difference which the number of the lunary months surpasses the number given of the solar months elapsed, during the same space of time; that thus in 228 solar months, which do make 19 years, there are 228 lunary months, and 7 months more, that is to say 235 lunary months. This therefore is a Period like to that of Numa and Mero, and to our Cycle of the golden number of 19 years, during which the Moon rejoyn'd it self 235 times to the Sun.
Yet in the sequel we shall see, that these Periods which accord together in the number of the lunar months and solar years, agree not in the number of the hours, by reason of the greatness of the solar year and of the lunar month, which is supposed various in these several Periods: and that the Indian is not subject to a fault so great, as the ancient Cycle of the Golden Number, which they have been obliged to expunge out of the Roman Kalender, in the Gregorian correction, because it gave the new Moons later than they are, almost a day in 312 years; whereas the New Moons determined by this Indian Period, agree with the true in this interval of time to near an hour, as will be found by comparing these Rules with the following.
II. / II.
1. Set down the Maasaken / --
2. Multiply it by 30 / --
3. Joyn thereunto the days of the current Month. / The months of the Moon are here reduced into days: but because they make all the months to consist of 30 days, there only will be some artificial months about 11 hours 16 minutes longer than the Astronomical, or some artificial days which begin at the New Moons, and are 22 minutes, 32 seconds shorter than the natural days of 24 hours, which begin always at the return of the Sun to the same Meridian.
4. Multiply the whole by 11. / --
5. Add thereunto also the number of 650. / They reduce the days into 11 parts, by multiplying them by 113 and they add thereto 650 elevenths, which do make 59 days and [illegible]. I find that these 59 days and 1/11 are the artificial days, which were elapsed to the day of the Epocha, since that an eleventh part of the natural day, and an eleventh of the artificial had began together under the meridian of the Indies, to which these Rules are accommodated.
6. Divide the whole by 703. / --
7. Keep the Numerator which you shall call Anamaan. / --
8. Take the quotient of the Fraction found Art. 6, and subtract it from the number found Art. 3. The remainder will be the Horoconne (that is to say, the number of the days of the AEra) which you shall keep. / Having laid apart what is always added by the 5th. Article, it appears by the 2d, 3d, 4th, 6th and 8th, operation, that as 703 is to 11, so the number of the artificial days, which results from the Operations of the 2d, and 3d. Art. is to the number of the days deducted to have the number of the natural days, which answers to this number of the artificial days: whence it appears, that by making the lunar month to consist of 30 artificial days, 703 of these days do surpass the number of the natural days, which equal them above eleven days.
One may find the greatness of the Lunar Month, which results from this Hypothesis: for if 703 Artificial Days do give an excess of 11 Days 30 of these Days which do make a Lunar Month, do give an excess of 163/303? in the Day; and as 703 is to 330, so 24 Hours are to 11 Hours, 15 Minutes, 57 Seconds; and deducting this Overpins from 30 Days, there remains 29 Days, 12 Hours, 44 Minutes, 3 Seconds for the Lunar Month, which agrees within a Second to the Lunar Month determined by our Astronomers.
As to the value of 59 Days and 1/11 which is added before the Division, it appears that if 703 Days do give 11 to subtract, 59 Days and 1/11 do give [x] in the Day, which do make 22 Hours, 11 Minutes and a half, by which the end of the Artificial Day, must arrive before the end of the Natural Day, which is taken for the Epocha.
The anamsan is the number of 703 part of the Day, which remain from the end of the Artificial Day, to the end of the current Natural Day. Use is made hereof in the sequel to calculate the motion of the Moon, as shall be afterwards explained.
The Quotient which is taken from the number of the Days found by the third Art. is the difference of the entire Days, which is found between the number of the Artificial Days, and the number of the Natural Days from the Epocha.
The Herotonne is the number of the Natural Days elapsed from the Astronomical Epocha to the current Day. It should seem that in rigour the Addition of the Days of the current Month, prescribed by the third Article, should not be made till after the Multiplication and Division, which serves to find the difference of the Artificial Days from the Natural, because that the Days of the Current Month are Natural, and not Artificial of 30 per mensem: but by the sequel it appears that this is done more exactly to have the Anamaan which serves for the calculation of the motion of the Moon.
III. / IIII.
1. Set down the Horoconne.
2. Divide it by 7.
3. The Numerator of the Fraction is the day of the Week. / It follows from this Operation and Advertisement, that if after the Division there remains 1, the current day will be a Sunday; and if nothing remains, it will be a Saturday: The Astronomical Epocha of the Horoconne is therefore a Saturday.
Note, That the first day of the Week is Sunday.
If it be known likewise what day of the Week is the day current, it will be seen whether the Precedent Operations have been well made.
IV. / IV.
1. Set down the Horoconne.
2. Multiply it by 800.
3. Subtract it by 373.
4. Divide it by 292207.
5. The Quotient will be the AEra, and the Numerator of the Fraction will be the Krommethiapponne, which you shall keep. / The days are here reduced into 800 parts. The number 373 of the third Article makes [x] of the day, which do make 11 hours and 11 minutes. They can proceed only from the difference of the Epochas, or from some correction, seeing that it is always the same number that is subtracted. The Epocha of this fourth Section may therefore be 11 hours and 11 minutes after the former.
The AEra will be a number of Periods of Days from this new Epocha, 800 of which will make 292207. The Question is to know what these Periods will be? 800 Gregorian Years, which very nearly approach as many Tropical Solar Years, do make 292194 Days. If then we suppose that the AEra be the number of the Tropical Solar Years from the Epocha, 800 of these Years will be 13 Days too long, according to the Gregorian correction.
But if we suppose that they are Anomalous Years, during which the Sun returns to his Apogeum, or Astral Years during which the Sun returns to the same fixt Star; there will be almost no error: for in 13 Days, which is the overplus of 800 of these Periods above 800 Gregorian Years, the Sun by its middle motion makes 12d. 48'. 48" which the Apogeum of the Sun does in 800 Years by reason of 57". 39'". per annum. Albategnius makes the Annual motion of the Sun's Apogeum 59". 4'". and that of the fix'd Stars 54". 34'". and there are some modern Astronomers which do make this annual motion of the Sun's Apogaeum 57". and that of the fix'd Stars 51"3. Therefore if what is here called AEra, is the number of the Anomalous or Astral Years: these Years will be almost conformable to those which are established by the antient and modern Astronomers. Nevertheless it appears by the following Rules, that they use this form of Year as if it were Tropical, during which the Sun returns to the same place of the Zodiack, and that it is not distinguished from the other two sorts of Years.
The Krommethiapponne which remains after the preceeding Division, that is to say, after having taken all the entire Years from the Epocha, will therefore be the 800 parts of the Day, which remain after the Sun's return to the same place of the Zodiack: and it appears by the following Operations that this place was the beginning of Aries. Thus according to this Hypothesis the Vernal middle AEquinox will happen 11 Hours 11' after the Epocha of the preceeding Section.
V. / V.
1. Set down the Krommethiapponne.
2. Subtract from it the AEra.
3. Divide the remainder by 2.
4. Neglecting the Fraction, subtract 1 from the quotient.
5. Divide the remainder by 7, the Fraction will give you the day of the Week.
Note, That when I shall say the Fraction, I mean only of the Numerator. / Seeing that in the third Art. the day of the week is found by the Horoconne, after a very easie manner, it is needless to stay on this which is longer and more compounded.
VI. / VI.
1. Horoconne.
2. Subtract from it 621.
3. Divide the remainder by 3232. The Fraction is called Outhiapponne, which you shall keep. / This Subtraction of 628, which is always deducted from the Horoconne, what number soever the Horoconne contains, denotes an Epocha, which is 621 days after the Epocha of the Horoconne.
The number 3232 must be the number of the Days, which the Moon's Apogaeum employs in running through the Circle of the Zodiack: 3232 Days to make 8 Indian Years and 310 Days. During that time this Apogaeum finishes a Revolution after the rate of 6'. 41". which it performs in a Day, even according to the Astronomers of Europe. The Apogaeum of the Moon does consequently finish its Revolution 621 days after the Epocha of the Horoconne. 'Tis here performed then; as 3232 days are to a Revolution of the Apogaeum, so the number of the days is to the number of the Revolutions of the Apogaeum. They keep the remainder which is the number of the days called Onthiapponne. The Outhiapponite will therefore be the number of days elapsed from the return of the Moon's Apogaeum to the beginning of the Zodiac; which will more evidently appear in the sequel.
If you would have the day of the Week by the Outhiappone, take the Zuotient of the aforesaid Division; multiply it by 5, then joyn it to the Outhiapponne, then subtract thence two days, divide it by 7, the Fraction will show the day.
Whatever is before is called Poulasouriat, as if one should say the Force of the Sun. / Having already explained the true method of finding the day of the Week, it is needless to stay here. Leaving the care of examining it, and searching the ground thereof, to those that shall have the curiosity.
Notwithstanding the name of the Sun's Force which is here given to the precedent Operations, it is certain that what has hitherto been explained, belongs not only to the Sun, but likewise to the Moon.
VII. / VII.
1. Set down the Krommethiapponne.
2. Divide it by 24350.
3. Keep the quotient, which will be the Raasi, that is to say, the Sign where the Sun will be. / To find what the number 24350 is; it is necessary to consider, that the Krommethiapponne are the 800 parts of the day which remains after the Sun's return to the same place of the Zodiac, and that the solar year contains 292207 of these parts, as has been declared in the explication of the fourth Section. The twelfth part of a year will therefore contain 24350 and 7/180? of these 800 parts: wherefore the number 2435- denotes the twelfth part of a solar year; during which the Sun by its middle motion makes a Sign.
Seeing then that [illegible] of a day do give a Sign, the Krommethiapponne divided by 24350 will give to the Quotient the Signs which the Sun has run since his return by his middle motion to the same place; The Raasi then is the number of the Signs; run through by the middle motion of the Sun. They here neglect the Fraction, 7/23? so that the solar year remains here of 292 [illegible], that is to say of 365 days 1/4?, like the Julian year.
4. Lay down the Fraction of the aforesaid Division, and divide it by 811.
5. The Quotient of the Division will be the Ongsaa, that is to say, the degree wherein the Sun will be. / Seeing that by the preceding Article 24 [illegible] of a day do give a Sign of the Sun's middle Motion, the 30th part of 24 150/ooo will give a degree which is the 30th part of a Sign. The 30th part of 24330 is 811; which do make a degree: dividing the remainder by 811 [illegible], they will have the degree of the Sun's middle motion. Here they neglect that ; which can make no considerable difference.
6. Set down the Fraction of this last Division, and divide it by 14.
7. The Quotient will be the Libedaa, this is to say the Minute.
8. Subtract 3 from the Libedaa.
9. Place what belongs to the Libedaa, underneath the Ongsaa, and the Ongsaa underneath the Raasi: This will make a Figure which shall be called the Mettejomme of the sun, which you shall keep. I suppose it is locus medius Solis. / Seeing that in a degree there are [illegible] parts; in a minute, which is the 60th part of a degree, there will be 13!! of these parts. Neglecting the Fraction, they take the number 14, which dividing the remainder, will give the minutes. The Subtraction which is here made of three minutes is a reduction whereof we shall speak in the sequel.
It is here prescribed to put the Degrees under the Signs, and the Minutes under the Degrees in this manner.
Raasi, Signs.
Ongsaa, Degrees.
Libedaa, Minutes.
This Disposition of the Signs, Degrees, and Minutes one under the other is called a Figure, and it here denotes the middle place of the Sun.
VIII. / VIII.
To find the true place of the Sun,
1. Set down the Mettejomme of the Sun, that is to say, the figure which comprehends what is in the Raasi, Ongsaa and Libedaa.
2. Subtract 2 from the Raasi. But if this cannot be, add 12 to the Raasi, to be able to do it, then do it.
3. Subtract 20 from the Ongsaa. But if this cannot be, deduct 1 from the Raasi, which will amount to 30 in the Ongsaa, then you shall deduct the aforesaid 20. / The number 2[?], which is subtracted from the Raasi, in the second Article, and the number 20 in the third Article, are 2 Signs and 20 degrees, which doubtless denotes the place of the Suns Apogaeum according to this Hypothesis; in which there is not seen any number which answers to the motion of the Apogaeum. It appears then that this Apogaeum is supposed fix'd to the 20th degree of Gemini, which precedes the true place of the Apogaeum, 25 it is at present 17 degrees, which this Apogaeum performs not in less than 1000 years, or thereabouts: From whence it may be judged that the Epocha of this method is about a thousand years before the present age. But as the greatness of the year agrees better here with the Suns return to the Apogaeum and the fixed Stars, than with the Suns return to the Equinoxes; it may be that the beginning of the Signs here used, is not at present in the Equinoxial point, but that it is advanced 17 or 18 degrees, and so it will be necessary to be corrected by the Anticipation of the Equinoxes. Here then they subtract the Suns Apogaeum from its middle placed called Mattejomme, to have the Suns Anomalia: and the number of the signs of this Anomalia is that which they call Kenne.
4. What will afterwards remain, shall be called Kenne. / It appeareth by these Rules that the Kanne is the number of the half-signs of the distance of the Apogaeum or Periganat [?], taken according to the succession of the Signs, according as the Sun is nearer one term than the other: So than in the 5th Article is taken the distance of the Apogaeum according to the succession of the Signs; in Article 6th the distance of the Perigaum, against the succession of the Signs: in Article 7th the distance of the Perigaum according to the succession of the Signs; and in Article 8th the distance of the Apogaeum, contrary to the succession of the Signs. In the 6th, 7th, and 8th Articles it seems, that it must always be understood. Multiply the Raasi by 2, as it appears in the sequel.
In the 6th Article when the degrees of the Anomalia exceed 15, they add 1 to the Kanne; because that the Kanne, which is a half Sign, amounts to 15 degrees.
The degrees and minutes of the Kanne are here reduced into minutes, the number of which is called the Ponchalit.
It appears by these Operations, that the Chaajaa is the AEquation of the Sun calculated from 15 to 15 degrees, the first number of which is 35, the second 67, the third 94; and that they are minutes, which are to one another as the Sinns of 15, 30, and 45 degrees from whence
It follows that the Equation of 60, 75, and 90 degrees are 116, 129, 134. / 35-67-94-116-129-134
which are set apart in this form, and do answer in order to the number of the Kanne, 1, 2, 3, 4, 5, 6.
As for the other degrees they take the proportional part of the difference of one number to the other, which answers to 15 degrees, which do make 900 minutes, making: as 900, to the difference of two Equations; so the minutes which are in the overplus of the Kanne, to the proportional part of the Equation, which it is necessary to add to the minutes which answer to the Kanne to make the total Equation. They reduce these minutes of the Equation into degrees and minutes, dividing them by 60. The greatest Equation of the Sun is here of 2 degrees, 12 min. The Alphonsine Tables do make it 2 degrees, 10 minutes: We find it of 1 degree, 57 minutes. They apply the Equation to the middle place of the Sun, to have its true place which is called Sommepont.
5. If the Kenne is 0, 1, or 2, multiply it by 2, you will have the Kanne.
6. If the Kenne is 3, 4, or 5; you shall subtract the figure from this figure 5-29-60, which is called Attathiat, and amounts to 6 Signs.
7. If the Kenne is 6, 7, 8; subtract 6 from the Raasi, the remainder will be the Kanne.
8. If the Kenne is 9, 10, 11; subtract the figure from this figure 11-29-60 which is called Touataasamounetonne, and amounts to 12 Signs: the remainder in the Raasi will be the Kanne.
9. If you can deduct 15 from the Ongsaa, add 1 to the Kanne, if you cannot, add nothing.
10. Multiply the Ongsaa by 60.
11. Add thereunto the Libedaa, this will be the Pouchalit, which you shall keep.
12. Consider the Kanne. If the Kanne is 0, take the first number of the Chaajaa of the Sun, which is 35; and multiply it by the Pouchalit.
13. If the Kanne is some other number, take according to the number, the number of the Chajaa aattit, and subtract it from the number underneath. Then what shall remain in the lower number, multiply by it the Pouchalit. As for example, if the Kanne is 1, subtract 35 from 67, and by the rest multiply. If the Kanne is 2, subtract 67 from 94, and by the rest multiply the Pouchalit.
14. Divide the Sum of the Pouchalit multiplied by 900.
15. Add the Quotient to the superior number of the Chajaa, which you have made use of.
16. Divide the Sum by 60.
17. The quotient will be Ongsaa, the Fraction will be the Libedaa. Put an 0 in the place of the Raasi.
18. Set the figure found by the preceding Article over against the Mattejomme of the Sun.
19. Consider the Ken aforesaid. If the Ken is 0, 1, 2, 3, 4, 5; It is called Ken subtracting: Thus you shall subtract the figure found in the 17 Article from the Mattejomme of the Sun.
20. If the Ken is 6, 7, 8, 9, 10, 11, it is called Ken additional: So you shall joyn the said figure to the Mattejomme of the Sun: which will give out at last the Sommepont of the Sun, which you shall precisely keep.
19. [21?] This Equation, conformably to the rule of our Astronomers in the first demi-circle of the Anomalia, is subtractive; and in the second demi-circle, additional. Here they perform the Arithmetical operations placing one under the other, what we place side-ways; and on the contrary, placing side-ways what we place one under the other. As for Example:
-- / The Mattejomme / The Chajaa / The Sommepont / --
Raasi / 8 / 0 / 8 / Signs
Ongsaa / 25 / 2 / 27 / Degrees
Libedaa / 40 / 4 / 44 / Minutes
-- / Middle Place / Equation / True Place / --
IX. / IX.
1. Set down the Sommepont of the Sun.
2. Multiply by 30 what is in the Raagi.
3. Add thereto what is in the Ongsaa.
4. Multiply the whole by 60.
5. Add thereunto what is in the Libedaa.
6. Divide the whole by 800, the Quotient will be the Reuc of the Sun.
7. Divide the remaining Fraction by 13, the Quotient will be the Naati reuc, which you shall keep underneath the Reuc. / It appears by these Operations that the Indians divide the Zodiac into 17 equal parts, which are each of 13 degrees, 40 minutes. For by the six first Operations the signs are reduced into degrees, and the minutes of the true place of the Sun into minutes; an din dividing them afterwards by 800, they are reduced into 27 parts of a Circle; for 800 minutes are the 27th part of 21600 minutes which are in the Circle, the number of the 27 parts of the Zodiack are therefore called Reuc, each of which consists of 800 minutes, that is to say, of 13 degrees, 40 minutes. This division is grounded upon the diurnal motion of the Moon, which is about 13 Degrees, 40 Minutes; as the division of the Zodiack unto 360 Degrees has for foundation the diurnal motion of the Sun in the Zodiack, which is near a Degree.
The 60 of these parts is 13-1/3; as it appears in dividing 800 by 60, wherefore they divide the Remainder by 13, neglecting the fraction, to have what is here called Nati-reuc, which are the Minutes or 60 parts of a Reuc.
X. / X.
For the Moon. To find the Mattejomme of the Moon.
1. Set down the Anamaan.
2. Divide it by 25.
3. Neglect the Fraction, and joyn the Quotient with the Anamaan.
4. Divide the whole by 60, the Quotient will be Ongsaa, the Fraction will be Libedaa, and you shall put an 0 to the Raasi. / According to the 7th Article of the III Section, the Anamaan is the number of the 703 parts of the day, which remain from the end of the Artificial day to the end of the Natural day. Altho according to this rule, the Anamaan can never amount to 703: yet if 703 be set down for the Anamaan, and it be divided by 25, according to the 2d Article, they have 28 [illegible], for the Quotient. Adding 28 to 703, according to the third Article, the sum 731 will be a number of minutes of a degree. Dividing 731 by 60, according to the fourth Article, the Quotient which is 12d. 11', is the middle diurnal motion, by which the Moon removes from the Sun.
From what has been said in the II Section, it results that in 30 days the Anamaan augments 330. Dividing 330 by 24, there is in the Quotient 13 [illegible]. Adding this Quotient to the Anamaan, the summ is 343, that is to say, 5 d., 43'. which the Moon removes from the Sun in 30 days, besides the entire Circle.
The European Tables do make the diurnal motion of 12d, 11'. and middle motion in 39 days, of 5d. 43'., 21", besides the entire Circle.
5. Set down as many days as you have before put to the month current. Sect. II. n. 3.
6. Multiply this number by 12.
7. Divide the whole by 30 the Quotient, put it to the Raasi of the preceding figure which has an 0 at the Raasi, and joyn the fraction to the Ongsaa of the figure.
8. Joyn this whole figure to the Mattejomme of the Sun.
9. Subtract 40 from the Libedaa. But if this cannot be, you may deduct 1 from the Ongsaa, which will be 60 Libedaa.
10. What shall remain in the figure is the Mattejomme of the Moon sought. / After having found out the degrees and the minutes which agree to the Anamaan, they seek the signs and degrees which agree to the Artificial days of the current month. For to multiply them by 12, and to divide them by 30, is the same thing as to say, If thirty Artificial days do give 12 Signs, what will the Artificial days of the current month give? they will have the Signs in the Quotient. The Fractions are the 30ths of a Sign, that is to say, of the degrees. They joyn them therefore to the degrees found by the Anamaan, which is the surplusage of the Natural days above the Artificial.
The Figure here treated of is the Moons distance from the Sun, after they have deducted 40 minutes, which is either a Correction made to the Epocha, or the reduction of one Meridian to another: as shall be explain'd in the sequel. This distance of the Moon from the Sun being added to the middle place of the Sun, gives the middle-place of the Moon.
XI. / XI.
1. Set down the Outhiappone.
2. Multiply by 3.
3. Divide by 808.
4. Put the Quotient to the Raasi.
5. Multiply the fraction by 30.
6. Divide it by 808, the Quotient will be Ongsaa.
7. Take the remaining fraction, and multiply it by 60.
8. Divide the summ by 808, the Quotient will be Libedaa.
9. Add 2 to the Libedaa; the Raasi, the Ongsaa, and the Libedaa will be the Mattejomme of Louthia, which you shall keep. / Upon the VI. Section it is remarked that the Onthiapponne is the number of Days after the return of the Moon's Apogaeum, which is performed in 3232 Days: 808 Days are therefore the fourth part of the time of the Revolution of the Moon's Apogaeum, during which it makes 3 Signs, which are the fourth part of the Circle.
By these Operations therefore they find the motion of the Moon's Apogaeum, making as 808 Days are to 3 Signs; so the time passed from the return of the Moon's Apogaeum is to the motion of the same apogaeum during this time. It appears by the following Operation that this motion is taken from the same Principle of the Zodiack, from whence the motion of the Sun is taken.
The Mattepomme of Louthia, is the Place of the Moon's Apogaeum.
XII. / XII.
For the Sommepont of the Moon.
1. Set down the Mattejomme of the Moon.
2. Over against it set the Mattejomme of Louthia.
3. Subtract the Mattejomme of Louthia from the Mattejomme of the Moon.
4. What remains in the Raasi will be the Kenne.
5. If the Kenne is 0, 1, 2, multiply it by 2, and it will be the Kanne.
6. If the Ken is 3, 4, 5, subtract it from this figure: 5 / 29 / 60.
7. If the Ken is 6, 7, 8, subtract from it 6.
8. If the Ken is 9, 10, 11, subtract it from this figure: 11 / 29 / 60.
9. If the Kenne is 1 or 2, multiply it by 23 this will be the Kanne.
10. Deduct 15 from the Ongsaa, if possible; you shall add 1 to the Raasi; if not, you shall not do it.
11. Multiply the Ongsaa by 60, and add thereunto the Lebidaa, and it will be the Pouchalit, that you shall keep.
12. Take into the Moons Chajaa the number conformable to the Kanne, as it has been said of the Sun; subtract the upper number from the lower.
13. Take the remainder, and therewith multiply the Pouchalit.
14. Divide this by 900.
15. Add this Quotient to the upper number of the Moons Chajaa.
16. Divide this by 60, the Quotient will be Ongsaa, the Fraction Libedaa, and an 0 for the Raasi.
17. Opposite to this figure set the Mattejomme of the Moon.
18. Consider the Ken. If the Ken is 0, 1, 2, 3, 4, 5, subtract the figure of the Moons Mettejomme; if the Ken is 6, 7, 8, 9, 10, 11, join the two figures together, and you will have the Sommepont of the Moon, which you shall keep. / All these Rules are conformable to those of the VIII. Section, to find the place of the Sun, and are sufficiently illustrated, by the explication made of that Section.
The difference in the Chajaa of the Moon, discoursed of in the 14th and 15th Article. This Chajaa consists in these numbers: 77 / 1_8 / 209 / 256 / 286 / 296.
The greatest Equation of the Moon is therefore of 4 degrees 56 minutes, as some Modern Astronomers do make it, though the generality do make it of 5 degrees in the Conjunctions and Oppositions.
XIII. / XIII.
Set down the Sommepont of the Moon, and operating as you have done in the Sommepont of the Sun, you will find the Reuc and Nattireuc of the Moon. / This Operations has been made for the Sun in the IX Section. It is to find the position of the Moon in her Stations, which are the 27 parts of the Zodiac.
XIV. / XIV.
1. Set down the Sommepont of the Moon.
2. Over against it set the Sommepont of the Sun.
3. Subtract the Sommepont of the Sun from the Sommepont of the Moon, and the Pianne will remain, which you shall keep. / The Pianne is therefore the Moon's distance from the Sun.
XV. / XV.
1. Take the Pianne and set it down.
2. Multiply the Raasi by 30, add the Ongsaa thereunto.
3. Multiply the whole by 60, and thereunto add the Libedaa.
4. Divide the whole by 720, the Quotient is called Itti, which you shall keep.
5. Divide the Fraction by 12, the quotient will be Natti itti.
The end of the Souriat. / These three first Operations do serve to reduce the Moon's distance from the Sun into minutes; dividing it by 720, it is reduced to the 30 part of a Circle, for 720 minutes are the 30th part of 21600 minutes, which do make the whole circumference. The ground of this division is the Moons diurnal motion from the Sun, which is near the 30th part of the whole Circle. They consider then the Position of the Moon, not only in the Signs and in her stations, but also in the 30th parts of the Zodiack, which do each consist of 12 degrees, and are called itti; dividing the remainder by 12, they have the minutes, or sixtieth parths of an itti, which do each consist of 12 minutes of degrees, which the Moon removes from the Sun in the sixtieth part of a day; these sixtieth parts are called natti itti.