Tibetan astronomy

The structure of the world according to Tibetan astronomy. The sizes of the sun and moon are not shown to scale.

The Tibetan Astronomy ( Tib. : Skar rtsis , "Calculation of the star (resorts)") is the historic Tibet popular science ( Tib. : Rig gnas ; "City of Knowledge") about the structure of the earth and the universe, about the Calendar calculation called calculation of the structure and the components of the Tibetan astronomical calendar , via the calculation of the movement of the ten planets known in Tibet including the sun , the moon , the lunar orbit nodes and the comet Encke and via the calculation of solar and lunar eclipses . The astronomical calculations were carried out with the Tibetan sand tobacco .

Tibetan astronomy and its calculations are still practiced today both in and outside of Tibet to create the annual calendar.

history

The astronomer Phugpa Lhündrub Gyatsho (1st half of the 15th century)

The astronomer Norsang Gyatsho (2nd half of the 15th century)

The regent and astronomer Sanggye Gyatsho

The Tibetan astronomy is of Indian origin, being on the astronomical teachings of the first chapter of Kālacakratantra based, an Indian tantric teaching text whose Sanskrit -Textversion may not have originated before 1027 and the first time in the second half of the 11th century, the Tibetan translated has been.

The astronomical calculations contained in the Kālacakratantra correspond to what was usually presented in numerous practical arithmetic books of Indian astronomy and calendar calculation. In Sanskrit , such arithmetic books are called Karaņa or later in Tibetan byed rtsis ("practical arithmetic"). The calculation parameters used have been shortened or rounded up to facilitate the practical implementation of the calculations in such plants.

The starting point for the development of a separate Tibetan astronomy was the thesis, which was widespread in the canonical text Vimalaprabhā, a commentary on the Kālacakratantra, that unbelievers had the true astronomy taught by the Buddha and recorded in a lost root tantra (Tib .: rtsa rgyud ) with malicious intent falsified and these falsifications spread in the first chapter of the Kālacakratantra.

The task of the Tibetan astronomers was thus ultimately understood by them as a “reconstruction of the true Siddhānta astronomy taught by Buddha”, which was referred to in Tibetan as grub rtsis . Observations of the starry sky were curiously of no significant importance for the development of this astronomy.

Various school traditions of astronomy developed in Tibet in the 15th century. The most famous of these were the Phug-pa school and the mTshur-phu school .

As a result of the intensive occupation with astronomy and calendar calculation, numerous works on astronomy were created in Tibet, the most important of which were published as block prints. In particular, the Vai insbesondereūrya dkar po of the regent Sanggye Gyatsho should be mentioned here .

Important for the development of Tibetan astronomy and calendar calculations personalities were among others drogön chögyal phagpa , Rinchen Buton Drup , Phugpa Lhundrub Gyatsho , Norzang Gyatsho , Pelgön Thrinle , Tshurphu Jamyang Chenpo Döndrub Öser , the regent Desi Sangye Gyatso, Karma Ngeleg Tendzin ( Tib. : Karma nges legs bstan 'dzin ) and the Nyingma scholar Lochen Dharmaśrī (1654–1717).

Structure of the earth and the starry sky

Simplified 3D representation of the Tibetan astronomical world view. The closed region below the Weltberg and the region shown as a head above the Weltberg have been omitted.

The earth is presented as a hemisphere. It consists of four spherical shells, which in terms of material consist of one of the elements air, fire, water and earth. Here the air forms the outer hemisphere, followed by fire, water and the core consisting of earth. The round world mountain (Tib .: ri rab , lhun po ) rises out of the flat cut surface of this hemisphere in the middle of the earth core .

Around the Weltberg the sun, moon and planets move on the mantle of a truncated cone. While the fixed stars are distributed over the entire celestial hemisphere, which rotates clockwise around the world mountain once a day driven by a driving wind, the sun, moon and planets also have their own motion (Tib .: rang 'gros ) that causes each other these celestial bodies also move counterclockwise around the world mountain at different speeds.

Ecliptic and angular dimensions

The Tibetan division of the ecliptic into 12 signs of the zodiac (red) and 27 lunar houses (blue)

The great circle created by the projection of the sun's apparent orbit over the course of a year on the celestial sphere is called the ecliptic . From the point of view of the earth , all planets including the moon move on it with slight deviations in the so-called latitude .

The first division of this great circle in Tibet is that into 12 signs of the zodiac , which in Tibetan were called khyim . For the astronomical calculations, they were counted and labeled with the numbers 0 to 11. The same applies to the division of the ecliptic into the so-called 27 lunar houses or lunar stations, which were called rgyu-skar in Tibetan and which were counted from 0 to 26, as shown in the blue color in the figure below.

The arc or angle unit of measurement rgyu skar was divided into 60 chu tshod "arc hours". The chu tshod were divided into 60 chu srang "minutes of arc". A chu srang consisted of 6 dbugs "bow breath", which in turn were divided into "parts" (Tib .: cha shas ), some of which were of different sizes. It is clear that with this system of angular measurements the ecliptical lengths of the sun, moon and planets could be determined very precisely.

Calculation methods in Tibetan astronomy

The astronomer Pelgön Thrinle (15th – 16th centuries) reckons with the sand abacus

The Tibetans' astronomical calculations were carried out on the sand abacus. The corresponding calculation instructions, which form the core of the Tibetan representations of astronomical calculations, resemble today's computer programs. The specialty is that linear equations for the calculation of the ecliptical lengths of planets, temporal quantities and the equations of the center of celestial bodies were formulated exclusively as program texts for the execution of calculations on the sand abacus. Solving mathematical problems by thinking in the program structures of the Sandabakus is a specialty of Tibetan mathematicians and astronomers.

The numerical values of a size specification that do not follow the decimal system are written one below the other on the sand abacus. The number noted at the respective place is always an integer in Tibetan astronomy or when calculating on the sand abacus. The places are always placed one above the other, for example for 3 rgyu-skar , 26 chu-tshod , 5 chu-srang and 4 dbugs :

3
26th
5
4th

In this example, the values are (from top to bottom) 27, 60, 60 and 6. They are not noted separately in Tibet.

In order to reproduce such numbers in a space-saving manner, the number sizes are noted below in square brackets separated by commas and the place values after them are given in round brackets separated by a slash. The above numerical value is given as [3.26.5.4] / (27.60,60.6). Generally speaking, below those usually five-digit numbers as [ , , , , ] / ( , , , , ) written with whole numbers and the points values. ${\ displaystyle a_ {1}}$ ${\ displaystyle a_ {2}}$ ${\ displaystyle a_ {3}}$ ${\ displaystyle a_ {4}}$ ${\ displaystyle a_ {5}}$ ${\ displaystyle st_ {1}}$ ${\ displaystyle st_ {2}}$ ${\ displaystyle st_ {3}}$ ${\ displaystyle st_ {4}}$ ${\ displaystyle st_ {5}}$ ${\ displaystyle a_ {n}}$ ${\ displaystyle st_ {n}}$

The result of dividing two whole numbers without a remainder and the remainder of this division is referred to as a special notation ${\ displaystyle \ lceil {\ frac {a} {b}} \ rceil}$ ${\ displaystyle \ lfloor {\ frac {a} {b}} \ rfloor}$

Time calculation

Tibetan calendar: Beginning of the 3rd Hor month in the Tibetan calendar from Lhasa for the water pig year 1923/24

For the Tibetan calendar , world events are chronologically arranged in cyclical structures. The time units of these cycles are usually defined by astronomical phenomena. The age of the world begins and ends with the meeting of all planets at the zero point of the ecliptic , the year results from the completion of the apparent rotation of the sun around the earth, the month describes the time span between two new moons and the calendar day is the natural day.

The almanac created with the astronomical calendar calculation is a lunisolar calendar.

At the center of the Tibetan calendar is the calculation of the date (lunar day) within a month, with which a natural day or weekday is counted within a month. Furthermore, the ecliptical lengths of the sun and moon as well as two other astrologically significant quantities are calculated for each day of the week, which are called byed pa (Sanskrit: karaṇa) and sbyor ba (Sanskrit: yoga). Since the calculations boil down to the determination of these five components, the calendar calculation is also referred to as yan lag lnga bsdus “summary of five components”, often abbreviated to lnga bsdus .

Mean orbital times of the sun and moon as well as the five planets Mercury, Venus, Mars, Jupiter and Saturn

Since the 15th century, Tibetan astronomers have been fascinated by the ability to calculate the mean sidereal orbital times (tib .: dkyil 'khor ) and the mean angular velocities (tib .: dus longs ) of the sun, moon and the five planets on the sandabacus. She was interested in the mathematical calculation of these quantities related to the three types of day ( zhag gsum ), namely zodiac day ( khyim zhag ), lunar day (tib .: tshes zhag ) and natural day ( nyin zhag ). For this reason, these calculations were also called zhag gsum rnam dbye “Analysis according to the three types of day.” Ultimately, it took the Tibetan astronomers several centuries to satisfactorily solve the arithmetic problems associated with calculations with multi-digit number systems.

The ratio between the mean zodiac day and the mean lunar day was fundamental for all calculations, namely

A = [1,2] / (-, 65) = , ${\ displaystyle 1 + {\ frac {2} {65}} = {\ frac {67} {65}}}$

and between mid lunar day and natural day, viz

B = 1 - [0,1,1] / (-, 64,707) = . ${\ displaystyle 1 - {\ frac {1 + {\ frac {1} {707}}} {64}}}$

The lengths of lunar day and zodiac day

Initially, the Tibetan astronomers were interested in the length of time d in the astronomical time units lunar day (tib .: tshes zhag ) and zodiac day (tib .: khyim zhag ).

Since, by definition, the length of a natural day is 21,600 breaths (tib .: dbugs ), which corresponds to [1,0,0,0,0] / (-, 60,60,6,707) d, this amount is multiplied by B = 1 - [0,1,1] / (-, 64,707) = . The result, [0,59,3,4,16] / (-, 60,60,6,707) d, is the length of the lunar day. ${\ displaystyle 1 - {\ frac {1 + {\ frac {1} {707}}} {64}}}$

To determine the length of the zodiac day, multiply this result by A = [1,2] / (-, 65) = . The result, [1,0,52,4,168, 50] / (-, 60,60,6,707,65) d, is the length of the zodiac day. ${\ displaystyle 1 + {\ frac {2} {65}}}$

Sidereal orbital period and angular velocity of the sun

Since, by definition, the period of revolution of the sun is ( ) = 360 zodiac days (Tib .: nyi ma'i khyim zhag dkyil'khor ), 360 • A results in the period of revolution of the sun in mean lunar days (Tib .: nyi ma'i tshes zhag dkyil 'khor ): ${\ displaystyle U_ {S, ZT}}$

${\ displaystyle U_ {S, LT}}$ = [371,4,36,5,7] / (-, 60,60,6,13).

Multiplying this result by B, calculating 360 • A • B, you get the period of revolution of the sun in natural days (Tib .: nyi ma'i nyin zhag dkyil 'khor ):

${\ displaystyle U_ {S, d}}$ = [365,16,14,1] / (-, 60,60,6,13,707) d.

This results in the amount of 365.2705 d as the length of the tropical year, which, in contrast to the actual length of the tropical year of approx. 365.2422, means a difference of 0.0283 d. In Tibet, this resulted in a shift in the mean beginning of the year of 28.3 d, i.e. almost a whole month in a thousand years. In fact, corresponding calculations show that the beginning of the year in Tibet, according to the calendar calculations based on the Kālacakratantra, fluctuated between January 1st and February 1st in the 11th century, while in the 20th century they almost invariably fell between February and March.

For the calculation of the mean angular velocity, i.e. the change in the mean length of the sun, based on the respective time units zodiac day, lunar day and natural day, Sanggye Gyatsho gives the general rule in his Vaiḍūrya dkar po , the angular measure for a full revolution , ds 27 rgyu skar (moon houses) or 27 • 60 = 1620 chu tshod , to be divided by the respective orbital period of a planet. Because of the arithmetic difficulties in performing this task on the sandabakus, Sanggye Gyatsho gives the arithmetic instructions for all three types of day. The amount is then calculated for the mean angular velocity of the sun per natural day (Tib .: nyi ma'i nyin zhag yul longs )

1620: = [0,4,26,0,93,156] / (27.60,60,6,149,209) moon houses. ${\ displaystyle U_ {S, d}}$

Mean angular velocity of the sun per lunar day (Tib .: nyi ma'i tshes zhag yul longs ):

1620: = [0,4,21,5,43] / (27.60,60.67). ${\ displaystyle U_ {S, LT}}$

If you multiply this amount by 30, you get the mean length of the arc that the sun travels in a synodic month:

[2,10,58,1.17] / (27.60,60,6,67).

Orbital time and angular velocity of the moon

After a synodic month, the moon reaches the position of the sun in the ecliptic. The Tibetan astronomers calculate the mean angular velocity of the moon per synodic month by adding the mean orbital arc that the sun covers in a synodic month to the angular measure of a full revolution (27 • 60 = 1620 chu tshod ). The result is divided by 30 and you get the angular velocity of the moon per lunar day (Tib .: zla ba'i tshes zhag yul longs ):

[0.58.21.5.43] / (27.60,60,6,67).

To calculate the period of revolution of the moon in lunar days (Tib .: zla ba'i tshes zhag dkyil 'khor ), one divides the angular measure of a full revolution by the angular velocity per lunar day and obtains the result

${\ displaystyle U_ {M, LT}}$ = [27,45,21,4,430] / (-, 60,60,6,869).

If you multiply this result by B, i.e. the above-mentioned conversion factor from lunar days to natural days, you get the orbit time of the moon in natural days:

${\ displaystyle U_ {M, d}}$ = [27, 279,37,1250] / (-, 869,64,1414) d.

This value for the orbital period of the moon corresponds to 27.32166 d and is very precise compared to the exact value 27.321674 d.

The Tibetan astronomers calculate the amount for the angular velocity of the moon per natural day

[0,59,17,3,95367] / (27,60,6,149209) moon houses.

Orbit times of the planets Mercury, Venus, Mars, Jupiter and Saturn

For the sidereal orbital periods of the 5 planets Mercury (tib .: lhag pa ), Venus (tib .: pa sangs ), Mars (tib .: mig dmar ), Jupiter (tib .: phur bu ) and Saturn (tib .: spen pa ) the corresponding values were taken from the Kālacakratantra and not changed in the following period. These are:

Mercury 87.97 d (exact value 87.969 d).
Venus 224.7 d (exact value 224.701 d).
Mars 687 d (Exact value 686.98 d).
Jupiter 4,332 d (Exact value 4,332.59 d)
Saturn 10,766 d (Exact value 10,759.21 d)

With the exception of an insignificant deviation for Saturn, these values are surprisingly accurate.

The conversion of these values into date days and zodiac days and the calculation of the corresponding angular velocities is carried out in the same way as the procedure described for the corresponding values of the sun.

Mean ecliptical lengths of the planets Mercury, Venus, Mars, Jupiter and Saturn

Number of natural days that have passed since the epoch

To calculate the mean length of a planet, the Kālacakratantra first determines the number of natural days that have passed since the epoch . To do this, multiply the number of the past lunar months L (J, M) by 30 and add the number of the past lunar days T. The latter is the date number used to count a weekday within a month. This gives the number of lunar days that have passed since the epoch. To convert this value into natural days, all schools of Tibetan astronomy use the conversion factor B = 1 - [0,1,1] / (-, 64,707) = . ${\ displaystyle 1 - {\ frac {1 + {\ frac {1} {707}}} {64}}}$

Later schools of astronomy add a certain amount D to the number of past lunar days in the subtrahend. This gives, multiplied by, the time of day at which the first lunar day of the relevant weekday begins. Since this amount is subtracted, one shifts with 1 - D • the beginning of the calculation of the number of past natural days exactly to the epoch, ie the beginning of the 1st weekday of the 1st month of the 1st year. The result ${\ displaystyle {\ frac {1 + {\ frac {1} {707}}} {64}}}$ ${\ displaystyle {\ frac {1 + {\ frac {1} {707}}} {64}}}$

Day number = ${\ displaystyle ((L (J, M) \ cdot 30 + T) - \ lceil ((L (J, M) \ cdot 30 + T + D) ({\ frac {1 + {\ frac {1} { 707}}} {64}}) \ rceil}$

is the so-called unclear (tib .: mi gsal ) number of natural days that have passed since the epoch (tib .: nyin zhag gi grangs or spyi zhag ). It is therefore unclear because it has the error that it does not take into account the changes in the lengths of lunar days caused by the irregularities in the orbit of the sun and moon. For this reason, add the day of the week from the initial value WA (m) and divide the result by 7. The result is compared with the current day of the week of the calendar and corrected for the day number calculated above by the difference between the current day of the week and the calculated day of the week.

The addition made later by the addition of D in the subtrahend, which does not occur in the Kālacakratantra, is hyper-correct in that it has no numerical effect because of the subsequent mathematical correction.

Initial values

For the planets, the initial values (Anf) given for the epoch, i.e. the deviations of the lengths of the celestial bodies from the zero point of the ecliptical orbit, are not given in terms of the angular dimensions of the lunar houses, but by the number of days (d) that the individual planet needed, in order to reach its length at the time of the epoch after passing through the zero point of the ecliptic. The epoch of the Kālacakratantra, i.e. the beginning of the Nag month of the year 806, is used here.

List of the initial values of the five planets:

Mercury: Anf (M) = -71.23 d,
Venus: Beginn (V) - 8.4 d,
Mars: Anf (M) 167 d,
Jupiter: Anf (J) -2600 d,
Saturn: Anf (S) -4820 d.

Mean lengths of the five planets

To the number of days determined above, you add the initial value for each planet and divide the result by the orbital time. The remainder of this division gives the number of days for each planet that have passed since the passage of the respective planet through the zero point of the ecliptic (Tib .: sgos zhag ):

${\ displaystyle \ lfloor {\ frac {day number + start} {U}} \ rfloor}$ .

Since this number behaves in exactly the same way as the mean length of a planet given in the angular measure in relation to the angular measure of the full orbit in the ecliptic (= 27), the angular measure for the mean length for each planet is:

${\ displaystyle {\ frac {\ lfloor {\ frac {day number + beginning} {U}} \ rfloor \ cdot 27} {U}}}$ .

The slow feet of the planets: sidereal midpoint equations

Tibetan table of the equations of the center of the five planets Mercury, Venus, Mars, Jupiter and Saturn

Since the planets, in Tibetan astronomy including the sun and moon, do not move on a circular orbit, but rather move approximately on an ellipse around the earth, their angular velocity varies. These different angular velocities mean that the actual ecliptical lengths of these celestial bodies differ from the calculated mean lengths. The mathematical formula that is used to calculate these deviations from the mean length is called the midpoint equations. In the graph, these equations each result in a trigonometric curve.

With regard to the moon and the sun, the equations of the mid-point have been presented in detail in the context of the Tibetan astronomical calendar calculation. See the equation of the moon and the equation of the sun .

Since the calculation of the midpoint equations of the remaining five planets, which are called slow feet (Tib .: dal rkang ), was carried out with the appropriate division of the ecliptic into twelve sections analogous to the calculations of the midpoint equation of the sun, it is sufficient here to calculate the lengths of the zero points of the anomalous revolutions and the interpolation tables.

Zero points, aphelion and perihelion of the anomalous orbits of the planets:

planet	Zero point of the anomalous orbit in lunar houses	Farthest point from the sun (aphelion) in moon houses	Closest point to the sun (perihelion) in moon houses	Closest point to the sun after Kālacakratantra in degrees	Actual Value (Perihelion) AD 500	Actual Value (Perihelion) AD 1900
Mercury	[16.30] / (27.60)	[23.15] / (27.60)	[9.45] / (27.60)	130 degrees	144.1 degrees	165.9 degrees
Venus	[6.0] / (27.60)	[12.45] / (27.60)	[26.15] / (27.60)	350 degrees	290.5 degrees	310.2 degrees
Mars	[9.30] / (27.60)	[16.15] / (27.60)	[2.45] / (27.60)	36.66 degrees	38.5 degrees	64.2 degrees
Jupiter	[12.0] / (27.60)	[18.45] / (27.60)	[5.15] / (27.60)	70 degrees	80.2 degrees	102.7 degrees
Saturn	[18.0] / (27.60)	[24.45] / (27.60)	[11.15] / (27.60)	150 degrees	153.7 degrees	181.1 degrees

Table for the equations of the center of the five planets:

	n: access numbers of the equations (Tib .: rkang ´dzin )	Mars: slope of the function or multiplier (Tib .: sgyur byed )	Mars: initial values of the functions (Tib .: rkang sdom )	Mercury: slope of the function or multiplier (Tib .: sgyur byed )	Mercury: initial values of the functions (Tib .: rkang sdom )	Jupiter: slope of the function or multiplier (Tib .: sgyur byed )	Jupiter: initial values of the functions (Tib .: rkang sdom )	Venus: slope of the function or multiplier (Tib .: sgyur byed )	Venus: initial values of the functions (Tib .: rkang sdom )	Saturn: slope of the function or multiplier (Tib .: sgyur byed )	Saturn: initial values of the functions (Tib .: rkang sdom )
First half of the equations (Tib .: snga rkang ): Amounts to be added.	1	25th	25th	10	10	11	11	5	5	22nd	22nd
First half of the equations (Tib .: snga rkang ): Amounts to be added.	2	18th	43	7th	17th	9	20th	4th	9	15th	37
First half of the equations (Tib .: snga rkang ): Amounts to be added.	3	7th	50	3	20th	3	23	1	10	6th	43
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	4th	-7	43	-3	17th	-3	20th	-1	9	-6	37
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	5	-18	25th	-7	10	-9	11	-4	5	-15	22nd
Second half of the equations (Tib .: phyi rkang ): Amounts to be subtracted.	6th	-25	0	-10	0	-11	0	-5	0	-22	0

The fast feet of the five planets: Calculating the geocentric deviations of the longitudes of the planets

Retrograde motion and loop orbit of an outer planet

Table for calculating the geocentric deviations of the ecliptical longitudes of the planets

Another correction of the mean length of a planet, which is calculated in Tibetan astronomy after taking the equations of the center point into account, is based on the fact that, for an observer on earth, the movement of the earth is mirrored in the apparent movement of the planets.

If you draw straight lines from the center of the earth's orbit through the planetary positions 1 to 7 in the figure on the right, angles result at the intersection points 1 to 7 of the planetary positions that describe the deviation of the geocentric length from the calculated sidereal length.

In Tibetan astronomy, these deviations are calculated using a system of linear equations called “quick feet” (Tib .: myur rkang ). Tibetan astronomical works contain tables with the initial values and slopes of these equations, which are used to calculate the deviation of geocentric longitudes from sidereal longitudes. The calculation method is based entirely on the information in the Kālacakratantra and has not been changed by the Tibetan astronomers. It can be assumed that the values used for the calculation were obtained purely empirically from longer observation series.

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(Sanskrit) Kālacakratantra. (Tibetan) mChog gi dang-po sangs-rgyas las phyung-ba rgyud kyi rgyal-po dus kyi 'khor-lo.
Grags-pa rgyal-mtshan: Dus-tshod bzung-ba'i rtsis-yig .
karma Nges-legs bstan-'dzin: gTsug-lag rtsis-rigs tshang-ma'i lag-len 'khrul-med mun-sel nyi-ma ñer-mkho'i' dod-pa 'jo-ba'i bum- bzang (block printing).
Phug-pa Lhun-grub rgya-mtsho: Legs par bshad pa padma dkar-po'i zhal gyi lung. Beijing 2002
sde-srid Sangs-rgyas rgya-mtsho: Phug-lugs rtsis kyi legs-bshad mkhas-pa'i mgul-rgyan vaidur dkar-po'i do-shal dpyod-ldan snying-nor (block print)
sde-srid Sangs-rgyas rgya-mtsho: bsTan-bcos vaiḍūrya dkar-po las dri-lan 'khrul-snang g.ya'-sel don gyi bzhin-ras ston-byed (block print)
Nag-dbang: sNgon med-pa'i bstan-bcos chen po vaiḍūrya dkar-po las' phros-pa'i snyan-sgron nyis-brgya brgyad-pa (block print)
Winfried Petri: Indo-Tibetan Astronomy. Habilitation thesis to obtain the venia legendi for the subject history of natural sciences at the high natural sciences faculty of the Ludwig Maximilians University in Munich . Munich 1966
Dieter Schuh : Studies on the history of the Tibetan calendar calculation . Steiner, Wiesbaden 1973 ( Directory of oriental manuscripts in Germany . Supplement 16, ZDB -ID 538341-9 ).
Dieter Schuh: Basics of the development of the Tibetan calendar calculation . Journal of the German Oriental Society, Supplement II. XVIII. German Orientalist Day from October 1st to 5th, 1972 in Lübeck. Lectures, pp. 554-566