The Use of Interpolation Techniques in Chinese Calendar

by Ang Tlan-Se (University of Malaya)

Calendar-making had been the main concern of the Chinese ruling dass throughout the centuries. Socially, for an agricultural economy as found in China, the promulgation of a reliable calendar meant the fulfilment of the needs of the agrarian community and the acceptance of it signified recognition of imperial authority. Politically, based on a unique concept that _the emperor was chosen and given a mandate by the heavens, the validity of a calendar issued by the imperial court meant the manifestation of heavenly sanction. Therefore, in the interest of state security as weil as the continuity of imperial power, the emperors of all ages could not afford to overlook the responsibility of calendar-making. Since the beginning of recorded history, Chinese emperors had subtly turned calendar-making into a monopoly and accorded the calendrical personnel with special privileges.

From the early times until its replacement by the Gregorian calendar in the twentieth century, the Chinese calendar was based on a system of !uni­solar reckoning. As both the astronomical periods of the synodic month and the tropical year are incommensurable, frequent readjustment of the system was inevitable. The whole history of Chinese calendar-making, as pointed out by Joseph Needham, is that of successive attempts to reconcile the irreconcilable 1•

In their efforts to produce a good system to reconcile the two fundamental astronomical cycles, calendar-experts of the various periods kept on im­proving the current computation techniques or developed their own ones. Interpolation (the determination of f(x) for a value of x between two tabular values) was a case in question. An attempt is now made to show the parallel development of the interpolation techniques and the revision of the Chinese calendar.

To begin with, it should be noted that in their devotion to keep a constant watch at the heavens, the Chinese astronomers did not feel the need to make their Observations conform with a geometrically perceived cosmic structure. Their approach was to represent faithfully the movements of the celestial bodies and to reduce their Observations as accurately as possible to algebraic relations.

As early as the fourth century B.C. the astronomer Shih Shen 111 knew that the rate of lunar motion varied, and that the moon diverged from the

1 Joseph NEEDHAM, Science and Civilisation in China, Vol. 3, Cambridge, England, 1959, p. 390.

135

ecliptic north and south 2 • In the secend century, the calendar expert Liu Hung devoted some twenty years of serious thought about this anomaly before he set out to compose bis Ch'ien-hsiang Calendar 131 3• He recognised the progressive motion of the moon's perigee and the retrograde motion of the moon's node. He employed this knowledge to calculate the time of true syzygy and also the true position of the moon at quadrature and syzygy.

Let f (n) be the position of the moon on the nth day from the perigee and ~ = f (n + 1) - f (n), then the position of tbe moon at the end of a fraction (s) of a day after the nth day is given by the following expression:

f (n + s) = f (a) + s6

where 0 < s < 1.

However, since the motion of the moon is not uniform throughout eadl day, the approximate value thus obtained by Liu Hung was far from being accu­rate.

By the sixth century with the availability of good armillary spheres in China, more accurate values for apparent celestial motion were obtained, and hence greater accuracy of calendrical calculations was expected. In 527 Chang Tzu-hsin 141 discovered the annual inequality of the sun's motion 4•

It is said that the discovery was the culmination of continuous observations with an armillary sphere for some thirty years on an island. The results of Chang's Observations were utilized by Liu Ch'o 151 (544-610) in his Huang­chi 181 calendar in 600 5• Based on Chang's discovery that the sun's motion was slow after the vernal equinox and fast after the autumnal equinox, Liu Cho divided the tropical year into four sections of unequal intervals, namely, from vernal equinox to summer solstice, from summer solstice to autumnal equinox, from autumnal equinox to winter solstice, and from winter solstice to vernal equinox. Eadl of the four unequal intervals was further sub-divided into six equal parts. Observed values of the sun's motion at eadl of the

! See K'ai-yüan chan ching (compiled by CH'O-T'AN Hsi-ta 121 in 729 A. D.), eh. 11, p. 3a. · ·

3 Chin Shu, po-na edition, eh. 17, 1b, ff. Liu Hung gave the technical terms used for the acceleration and retarding phases of the anomalaus motion, chi and ch'ih respectively. The equations were additive during the former and subtrative during the latter. Eaeh phase was divided into two halves, a beginning (ch'u) an an ending (mo).

4 Sui shu, po na edition, eh. 17, p. 4a; Hsin T'ang shu, po na edition, eh. 27 (B) p. 1a; Y üan shih, po na edition , eh. 52, p . 28b.

5 Sui Shu, eh. 18, pp. 1a, ff. LI Yen [7), Chung suan chia ti nei ch'a Ia yen chiu (A Study of the interpolation formulae of Chinese mathematicians), Peking, 1957, pp. 21- 35; CH'IE Pao-tsung 181, Chung kuo shu hsüeh shih (History of Chinese mathe­matics) , Peking, 1964, pp. 102-103; YEN Tun-ehieb 191, "Chung suan chia ti chao ch'a shu" (Method of finite differences by Chinese mathematicians) , Shu hsüeh t'ung pao, Peking, 1955, no. 1, pp. 4--8. NAKAYAMA Shigeru in bis review of the nature of Chinese calendar says that "Liu Ch'o made the first recorded distinction between true (ting . ch'i 1101 and mean (p'ing ch'i 1111) eh'i intervals. See NAKAYAMA, A History of Japanese astronomy , Harvard, 1969, p. 68.

c 2 J .m ~ it : 00 JC r5 *~ C 3 J IIJ ~ : ~ * .1! C s J IIJ ~ c 6 J ~ ~ c 1 J *{I C a J ~. ~ ( 10) ')E~ ( 11 ) ~~

136

(4J5&T-rn c 9)M~~

twenty-four intervals were tabulated. Using these tabulated values, Liu Ch' o devised his interpolation technique to compute the position of the sun at any point between two intervals. His technique can be summarized as follows:

Let f (t) be the position of the sun along its ecliptic at time t and let t = ni where n = 1, 2, 3 . . . 24, and l = the length of one ch'i in days. Assuming that the observed values of the position of the sun at time nl, ni + .e, ni + 2l . .. be f(nl), f(n i + i), f(ni + 2l) ... The aim now is to find an expression tor t lt + s) where s is the number of days elapsed since the initial ch'i.

Letf(ni + i)-f(ni) = 6 1

f(ni + 2i) - f (ni + i) = 6 2

Then Liu Ch'o's formula is given by

f (ni + s) = f (nl) + + 6 1 : 62 s + -l-(61 - 62) -

Liu Ch'o·s formula can be explained geometrically as follows:

Figure 1

A . M ~ C 4----- L----~

1 J,

f

137

+

Let AC = CE = l, AM= s G and I are mid-points of AC and CE respectively.

Area of ACDB = 61 Area of CEFD = 6 2 All lines are either vertical or horizontal as they appear except BF. From areas of trapezia ACDB, CEFD and AEFB, we have

GH = ~ IJ = ~ and CD = 6 1 + 6 2

l , l 2l

Since 6 BLD = 6 HKJ, LB = KH = GH- IJ = 6 1-_; 6 2

AB = CD + LB = 6 1 + 62 + 2l

By properties of similar triangles,

AM s PB = AC. · LB = -l2- (6 t- 62)

MN = AB _ PB = 6 1 + 6 2 + 6 1 + 6 2 2t l

1 . Area of trapezium AMNB = -

2- (AB + MN). AM

= _1_ [ 6 1 + 6 2 + 2 2l

Liu Ch'o did not name his interpolation technique neither did he show detailed derivation of the expression given above. The whole procedure was paraphased in rhetoricallanguage. Nevertheless, he did give specific names to the three terms in the expression. Perhaps an analysis of the three terms contained in the expression may throw some light on the rationale behind Liu Ch'o·s technique.

138

Figure 2

8 V

A M c +----J.

From the above figure, it i5 5een that area of LTUD = 6 1-62

By proportion, area of L WMA = + (61 - 6 2). Thi5 i5 the 5econd term

in the expre55ion.

Area of ACDL

5 6t + 62 . . By proportion, area of L WMA = -

1- •

2 . Thl5 15 the fir5t term

in the expre55ion.

6 DUF and 6 BLD are congruent triangle5, DU = BL = LT. Hence, the area of L WXT = area of L WVB.

Adding the fir5t and 5econd term5 give the area AMVB. 1

Area of 6 BVN -2-BV.VN

VN PB

5 From Figure 1, it ha5 been shown that PB= AB- MN = ----p-

(.61- .6z).

Therefore, 6 BVN = + 5 · +(61 - 6 2). Thi5 i5 the third term.

Area of AMNB = Area of AMVB- 6 BVN

5 .61 + 6z 5 52

-~-· 2 +-~-(61-6z)--p(6t-.6t) ·

139

Liu Ch'o called the first term Ch'i End Rate 1121, the second term Aggregate Difference 1131 and the third term Separate Difference 1141. It is seen that if the acceleration or retardation of the sun's motion is uniform between t = ni and t = nl + U , then it i;; justifiable that f (t) is of degree 2. In reality, the sun does not move with uniform acceleration within any interval of time, f (t) is not of degree 2. Therefore, Liu Ch'o·s formula can only give the approximate value of f(t) . On account of the inequality of the sun's motion Liu Ch'o used two sets of average values for the Ch'i intervals in his com­putation. For the ch'i after the autumnal equinox, he assumed the average

, 16x10 , length of a eh i to be

11 Jays and for the eh i after the vernal equinox,

17x10 . , the average length was

11 days. Lm Ch o also used the same formula for

computing the motion of the moon, in whidl case f equals the length of a day. For all computational purposes f (nl) was tabulated and interpolation was treated as a problern involving equal intervals.

Suppose the numerical values of f (x) are given for values where the argument x increases by equal steps h.

f1-fo = 6, fo f1 = fo + 6. fo

This may be written as (1 + 6,) f 0 where 6. is an operator acting on f0 •

f2- f1 6 f1 f2 = f1 + 6 f, = fo + 6 fo + 6 fo + 6 2fo

= fo + 2 6. fo + 6 2fo which may be written f2 = (1 + 6.) 2fo.

f3 - f 2 6. f 2

fs 6 f2 fo + 2 6 fo + 6 2fo + 6 fo + 2 6. 2fo + 6. 3fo fo + 3 6 fo + 3 6, 2f0 + 6_ 3f0

(1 + 6 ) 3fo

Continuing in this manner fn may be expressed in the form fn = (1 + 6,) nf0 •

Using the bionomial theorem to expand this gives the Gregory-Newton formula of interpolation.

f = f + n "' f + n (n-1) "' !f + n (n-1) (n-2) A llf + n o Do 21 Do 31 uo

Sudl a formula was first discovered by James Gregory in 1670 and devel­oped into its modern form by Newton around the year 1676. Liu Ch'o's formula can be rearranged to show its identity with the Gregory-Newton formula as follows:

When i = 1, Liu Ch'o·s formula is in the form

f( +) 6.t+62 S2

n s = f(n) + s 2

+ s (61 - 6.2) --2-(6.t- 6t)

- s (6t ;- 6!) ]

( 12) *$ ( 13) *l~ (14) 7.JU~

140

f (n) + s [ 6n +

f (n) + s [ Llt +

= f (n) + s Llt +

(61- 62) -2 s (6t- 62)]

(s -;-l) (6z- 61)]

s (s -1) "' 2 2 Dt•

It is seen that if the third and higher order differences are neglected, the Gregory-Newton formula is exactly the same as Liu Ch'o·s formula.

Owing to political intrigues, Liu Ch'o's calendar was never put into official use 6, Some sixty-four years later, when Li Ch'un-fEmg 1151 was commissioned to compile his Lin-te 1161 calendar, he adopted Liu Ch'o's interpolation tech­nique for computing the position of the sun along the ecliptic 7• As both Liu Ch'o and Li Ch un-feng did not understand fully the inequality of the sun's motion, inadequacy of the calendar was inevitable.

As has been pointed out earlier on, in ancient and medieval China, the strong psychological sense of the correspondence between celestial and terrestrial reality played a crucial part in practical politics. Chinese em­perors would view with great alarm untoward celestial events such as the eclipses of the sun. By making the events predictable would relieve the emperors of their anxiety. Therefore, much premium was placed on the accuracy of the calendar in use.

One of the acid tests of the accuracy of a calendar was the eclipses it predicted. For example, during the ninth year of the K'ai-yüan period (721) when the Lin-te calendar failed to tally with eclipses of the sun, the emperor Hsüan-tsung passed a rescript, ordering the monk I-Hsing 1171 to make a new calendar 8• That the Lin-te calendar was loosing its accuracy in eclipse pre­diction was understandable. The most obvious reason was the employment of the equal interval interpolation formula for the treatment of the sun's motion on the ecliptic. I-Hsing recognized the validity of Liu Ch'o's formula but questioned its application in the computation of the sun's motion. He pointed out that the change in the speed of the sun' s motion is a gradual process. Thus, according to his reckoning, the inequality of the sun's motion can almost be repr.esented by the sine curve. Undoubtedly, I-Hsing had established the concept of the true ch'i periods. By this it is meant that the twenty-four ch'i periods were marked .by unequal intervals. Hence, Liu Ch'o's equal interval interpolation formula was found unsuitable for calculating the speed of the sun's motion between two contiguous ch'i. On this basis, I-Hsing developed Liu Ch'o's formula into one for unequal intervals. He used this formula in his Ta-yen 1181 calendar system.

In order to have a clear understanding of I-Hsing's formula in particular and the traditional method of presentation in general , the following is a full translation of the account for computing f (n + s) between two ch'i as given in the Hsin T'ang shu (Official History of the T'ang dynasty) 11 •

6 Sui shu, eh. 18, p. 3b. 7 Chiu T'ang shu, po na ~dition, eh. 32, p. 2a. For a biography of Li Ch'ung-feng,

see Chiu Tang shu eh. 79 and Hsin T'ang shu, eh. 204. 8 Hsin T'ang shu, eh. 27 (A), p. ta. • Hsin T'ang shu, eh. 28 (A), p. 7a & b.

c1s) * i'$Jfi. ( 16) •~ ( 17) -fr c1sJ * m 141

uTake the Expansion/Contraction Parts 1191 of the entry ch'i (,6 1) and add them to those of the following ch'i (,62). Multiply the sum (.61 + ,62) by twice the Hexagram Factor 6 1201 (s) and divided it by the sum of the Double

Hour Constants 1211 of the two ch'i (.l1, .l2). The result (s · ~: : t 2 ) is called

the End Rate 1221. Arrange again the Expansion/Contraction Parts of the two ch'i and multiply them separately by twice the Hexagram Factor 6. Divide the products thus obtained by their respective Double Hour Constants. Sub-

. [ .6t .6z ) ] tract the lesser from the greater. The remamder s · (-1-1- - -z;- is

called the Ch'i Difference 1231. In the case of ch'i after the solstices, add the Ch'i Difference to the End Rate; andin the case of ch'i after the equinoxes,

subtract the Ch'i Difference from the End Rate. The result [ s · ~: ! tz + s ( ~11 - ~2

) ] is the Beginning Rate 1241. Double the Ch'i Differ-

ence can multiply it by twice the Hexagram Factor 6 (s). Divide it again

by the sum of the Double Hour Constants of the two ch'i. The result [ 252

Lt + lz

( ~1 - ~~ ) ] is called the Daily Difference 1251. Halve it and add it to

or subtract it from the Beginning Rate. The result [ s · ~:: t2

+s(~ _ ~) + _1_ , 2s2 (~ _ ~) ] Lt Lz 2 Lt + l2 Lt lz

is called the True Rate 1261. Take the Daily Difference and substract it from (if the ch'i is after the solstices) or add it to (if the ch'i is after the equinoxes) the True Rate. The result

[ 5 • .6t+.62+s(~-~)- 52·(~-~)] lt + lz .l1 lz lt + lz lt Lz

yields the daily Expansion/Contraction Parts. The parts are gradually accu­mulated and are ad~ed to <;>r subtract from the Leading/Following Valuesf271 [f (t)] listed under the day of the entry ch'i to give the true value ofthat day [f (t + s)]."

The above rhetorical statement can be summerised in the following ex­pression:

142

( 20) 7\ ~ (25) 8 ~

c2o I& rx (26) Ji::$

(22) * $ (23) *' # (27) )t ~-

I-Hsing's formula can be explained geometrically as follows:

A M ~----

(i

1,

Figure 3

C C' J E 1" --------').

The points A, B, C, D, E, F, G, H, I, J, K, M, N and P are as defined in Figure 1. Note, however, the following modifications:

AC= l 1, CE = l 2 .

AC' = C'E, BD' = D'F LD' instead of LD in Figure I. Area of ACDB = ~1 Area of CEFD = 6t GH= ~ IJ = ~

ft I l2 1

AC' = GI = -2-(L1 + L2)

C'D' = 6t + ~t ft + l!

LB = KH = ~ - __fu_(6 LD'B= 6 KJH) ft ft

AB=C'D'+LB= 6.,+~2 + ~-~ L1 + Lt i 1 i2

AM 2s 6t 6t PB = AO: ·LB = L

1 + i2 (-l-1- - ~)

MN = AB _ PB 6., + 6.2 + 61 6t 21 + Lt -i-1- ~

2s

143

1 Area of trapezium AMNB = -

2- (AB + MN) · AM

=[ ~h + 62 + ~ ~ s ( ~ lt+l2 lt l! lt+l2 lt If f (t) is given, then

6, + 62 + 6, f (t + s) = f (t) + s · ---- s (--i1 + t2 t,

The above three terms can be given meaningful geometrical interpretation as follows:

Figure 4

I

.-s -.I A M G c: c I ]

LA C'D' = 6, + 62 t;+i;-

LB 6t 62 1;- --p;-

VN PB 2s ,~ _ili_)

lt + l! 11 12 AM s

61 + 62 f It is clear that the first term s · 11

+ l2

amounts to the area o rect.

AMWL while the second term s ( ~11 - ~22 ) amounts to the area of

rect. L WVB. The third term represents the area of the triangle VBN. This has tobe subtracted from the sum of the first two terms in order to obtain the required area AMNB.

144

The application of 1-Hsing's formula in the computation of the motion of the sun is, indeed, a clear improvement of the previous foz:mula used by Liu Ch'o. A comparison of this formula with Newton's general torward inter­polation formula brings to light their identity. Newton's formula for arbitrary intervals can be expressed as follows: f (t) = f (t0 ) + (t - t0 ) ~ (to, t

1)

+ (t- to) (t- t1} ~ (to, t1, t2} + ... where l; (to, tt), l; (to, t1, t2) denote the first divided difference and second divided difference respectively. We shall confine ourselves up to the second divided difference and let those of higher orders be zero.

If our time-line for I-Hsing's formula be defined as in Figure 5, we can · show that 1-Hsing's formula is identical with f (t} above.

Figure 5

+----11----t +---/2 ---4

A+- s -+M C E to t lt = (to + f ,) t2 = (to + 11 + 12)

From the above diagram we see that (t-to) = s, (t-t1} = s-11. By definition of divided difference, we have

f (to + /1} -f (to) 6.1 (to + i1}- to = _i_1_

6t t;-

6.1 _1_1_

145

This is exactly the same as the expression given by I-Hsing. Towards the end of the T'ang dynasty, Hsü Ang 1281 modified I-Hsing's

technique in his Hsüan-ming 1281 calendar of 822 10• From the position of the sun at the time of the true ch'i, Hsü Ang computed the longitude of the sun at any required time by using the formula:

61 si1 ( ~ f (t + s} = f (t} + s • --;-

1 +

.t i1 + iz i1

Hsü Ang's formula is basically the same as that given by I-Hsing. It is seen that the sum of the first and second terms in Hsü Ang's formula is equal to the sun of corresponding terms in I-Hsing's formula, that is,

__ili_ + .el ( __ili_ - _fu_) = 6t + 62 + (~-~) i1 i1 + i2 i1 i2 i1 + i2 it iz

Apart from modifying I-Hsing's formula for unequal intervals, Hsü Ang also modified Liu Cho's formula for equal intervals making it simple and identical with Newton-Gregory formula. Hsü Ang used an expression for computing the longitude of the moon, (n + s) days after the perigee as:

s s2 f(n + s} = f(n} + s6t + - 2-(61- 6z) --2- (~~- 6z).

If the formula is rewritten in the form: s (s-1)

f(n + s) = f(n) + s6f(n) + 2

6 2 f(n)

then it is exactly the same as the Newton-Gregory formula with 6 3 f (n) = 0. During the whole period of the Sung dynasty that followed, the calendar

experts merely followed the interpolation techniques handed down to them. It was not until the Yüan dynasty in the thirteenth century that the inter­polation formula was extended to include the third order difference. It has been noted earlier on, since the sixth century, calendar-experts who assumed uniform acceleration or retardation of the speed of the sun, moon and the five planets within a short period of time invariably used Liu Ch'o's formula in their computation. A break-through was made by Kuo Shou-ching 1321 and bis co-workers when they devised the Sbou-shih 1331 calendar system of 1281 11 • Kuo Shou-ching and his co-workers recognised that the speed of the sun, moon and the five planets was not the function of time of the first degree but of the second degree and that the total angular motion within a

10 Hsin T'ang shu, <h. 30 (A), p . la. After the Hsüan-ming calendar, the Ch'ung­hsüan 1301 calendar by PIE Kang [311 of 892 also followed the Ta-yen syste~.- See Hsin T'ang shu, <h. 30 (B), pp. la & b. For discussion of the interpolation tedmique during the Sui and T'ang periods, see YABUUCHI Kiyoshi, Zuitö rekihö shi no kenkyu (Researches in the history of calendrical science during the Sui and T'ang periods), Tokyo, 1944, pp. 70-73 and pp. 77-79.

u Yüan shih, eh. 52-55.

c 2s J ~~ (29J'RaJ1 C3oJ *~ (3IJ~Wd (32J~i~~ (33)~~ 146

certain intervalwas not the function of time of the second degree but of the third degree. By this the third order difference was instituted in Liu ch·o·s · formula.

However, in the Yüan-shih (History of the Yüan dynasty) where the Shou­shih calendar system is included, the interpolation tedmiques are not discribed in full . Details of these tedmiques are incorporated in the Ming­shih (History of the Ming dynasty) 12• Thus, by comparing the versions in both the dynastic histories, one can see the full employment of the inter­polation tedmiques in the computation of the non-uniform motion of the sun, moon and the planets . For the purpose of illustration only the case of the cal­culation of the irregularities of the sun's angular motion is attempted here.

In the solar inequality the sun's path is divided at the solstices and equi­noxes into four quarants of mean extent 91.31 (Chinese) degrees. (A round of the heavens is taken to be 365.25 degrees.) The inequality is symmetrical about the solstices, for the perigree and apogee of the sun's orbit were not distinguished from the winter and summer solstices respectively. Accroding to the constants given in the Shou-shih calendar system the sun takes 88.91 days to traverse ea<h of the two quadrants before and after the winter solstice, and 93.71 days to traverse each of the two quadrants before and after the summer solstice. Take the quadrant from the winter solstice to the vernal equinox as an example. If the mean motion is one degree per day, the sun has travelled 2.4 degrees in excess of the mean motion for the interval of 88.91 days. The Shou-shih calendar system divided the interval into six sub-divisions (0, l, 2l , 3l ... 6i) of 14.82 days ea<h. Observations of the true motion of the sun at ea<h of these six points were made. Then by sub­tracting these observed values from the values of the mean motion, the excess f (nl), expressed in degree parts, at ea<h intervalwas obtained. From these excess values, the divided difference up to the third order for ea<h interval of 14.82 days could be tabulated. From the data given it is noted that the third divided differences for each interval are equal while the fourth divided differences are all equal to zero. Hence, it is known that, in the Shou­shih calendar system, f (ni) was considered to be the function of l of degree three. However, in order to simplify the procedure by reducing l = 14.82

f (nl) to l = I, Kuo Shou-ching and his co-workers considered nt and called

this the Daily Mean Difference (341 (F (nl)]. By this consideration, the divided differences .were curtailed to only the first two orders. In the following table, all 'differences' are expressed in degree parts, one degree being ten thous­andparts.

Accu- Accumulated DailyMean 1st 2nd mulated Diffe rence Difference Difference Diiference

Days (nl ) f (n l) f (n.l) / nl Öt Ö t = F(nl )

1 14.82 7058.0250 476.25 -38.45 -1.38

21 29.64 12976.3920 437.80 -39.83 -1.38

31 44.46 17693.7462 397.97 -41 .21 -1.38

41 59.28 21148.7328 356.76 -42.59 -1.38

51 74.10 23279.9970 314.17 -43.97

61 88.92 24026.1840 270.20

l! Ming snih, eh. 34. ( 34) 8 zp.~ 147

From the above table it can be deduced that the second divided difference (6 2) at winter solstice is equal to -1.38, the first divided difference {6 1) = -38.45-(-1.38) = -37.07. The Daily Mean Difference [F {nl)] at winter solstice = 476.25 + 37.07 = 513.32. Thus, by the application of the inter-

f (nl) 1 polation formula F (nl) = ~ = 513.32 + n (-37.07) + - 2- (n - 1)

(-1.38). X

Let x be the days after the winter solstice, then 14.82 is taken as the

interval after the winter solstice. By employing the expression given for F (x), the Daily Mean, Difference at xth day is:

X 1 X X 14.82 ( 14.82 - 1) X 1.38 513.32 l4.tl2 X 37.07- -2-

37.07-0.69 0.69 14.82 X- (14.82) 513.32

513.32 2.64 x- 0.0031 x 2

Therefore, the Accumulated Difference for x days {x ~ 88.91 days)after the winter solstice is : f (x) = x.F (x) = 513.32x- 2.46x2 - 0.0031 x 3•

The coefficients of x, x2, x3 in the Shou-shih calendar system are known as True Difference 1351, Square Difference 1381 and Cubic Difference 1371 respec­tively . In this case both the coefficients of x 2 and x3 are negative. Generally the expression can take the form f (x) = ax + bx2 + cx3 . In the Yüan-shih, the general method for finding the Expansion/Contraction Difference 1381 (i. e. Equation of Centre) is given as follows 13 :

"Set up the Cubic Difference, multiply it by the Beginning/End Extent (seJ

(i. e. the nurober of days from the synodic phenomenon being considered to the solstice nearest it in space, not necessarily in time), and add the Square Difference, again multiplying by the Beginning/End Extent. The result is subtracted from the True Difference. The remainder is again multiplied by the Beginning/End Extent. The final result is the Expansion/Contraction Difference."

In the Shou-shih calender system, two sets of constants were given for two separate cubic equations. There were:

y = 0.051332x- 0.000246x2 - 0.00000031 x2

y = 0.048706 x' - 0.000221 x' 2 - 0.00000027 x'3

where y is the equation of centre expressed in Chinese degrees parts (1 degree = 100,000,000 parts), x the nurober of days counted from the perigee (writer solstice) , and x' the nurober of days counted from the apogee (summer solstice).

13 Yuan shih, d1. 54, pp. 4b- Sa.

(35) ~ ~ (36) .zp. ~ (37) lL ~ (38) /SI.~ ( 39JW *15N 148

In the same section, the .Yüan-shih mentions another method for the same purpose. lt says: "Set up day parts in the argument for the current quadrant and multiply by the Expansion!Contraction Parts for the day in question. Reduce by 10000 to get parts, and add to the listed Expansion/Con­traction Accumulation 1401. Reduce by 10000 to get degrees; the remainder is parts. This also yields Expansion/Contraction Difference. "

This ready-reckoning method avoids the relatively complex procedure just given by substituting simple linear interpolation from a table provided in the text. The current version of the Yüan-shih does not give the table. In the Ming-shih there is such a table whidl provides a day-by-day Iist of values for inequality parts (Expansion/Contraction Accumulation) and suc­cessive differences between these values (Expansion/Contraction Parts) 14•

A table of the day-by-day values for inequality parts counted from the perigee (winter solstice) could be constructed from the cubic equation for­mulated above. The first three divided differences were called Expansion Added Parts 1411, Square-Cubic Difference 1421 and Cubic Difference tuJ re­spectively. Let the first three differences be a, ß and y, then the Expansion Accumulation for x days after the winter solstice is:

ax- bx2 - cx3 x (x- 1) + ax+ß· 1·2

x (x - 1) (x - 2) y . 1. 2. 3

(a-+ + +) x + (-ß-2

_Y_) xt + _Y_xs 3 6

ß It can be seen that a = a - - 2- + y

3

b _ß_

2

c y

-6-

y - 6c

ß - 2b-6c =

a a-b-c=

+ _Y_ 3

-6x0.0031 -0.0186 .

2 x2 · 46-0.0186 4.9386 513.32-2.46-0.0031 510.8569

In fact from the general equation of f (x) = ax + bxt + cxa, it can be seen that when x = 1, 2, 3, 4 ... , the coefficients of the first term in f (x)

u Ming shih, ibid.

149

will have the corresponding increase by 1, 2, 3, 4 ... , while those in the second and third terms will increase correspondingly by their square and cube respectively. That is:

f (0) 0 f(l) a + b + c f (2) 2a + 2b + Be f(3) 3a + 9b + 27c f (4) 4a + 16b + 64c

Thus, a table of the daily accumulated difference and its orders of differ-ences can be constructed as follow:

Accumulated 1st 2nd 3rd 4th Difference Differ- Differ- Differ- Differ-

ence ence ence ence

Beginning Day f (0) = 0

First a+b+c

Day f (1) = a + b + c 2b + 6c

Second a + 3a + 7c 6c

Day f (2) = 2a + 4b + Be 2b + 12c 0

Third a +Sb+ 19c 6c

Day f (3) = 3a + 9b + 27c 2b + lBc 0

Fourth a + 7b + 37c 6c

Day f(4) = 4a + 16b + 64c 0

Needless to say, in incorporating the sun's anomalolis motion in their Shou-shih calendar system, Kuo Shou-dling and his co-workers had devel­oped Liu Cho's equal interval interpolation formula further by adding the third term containing the third order differences to improve the approxima­tion. Assuming the fourth order difference to be zero, their formula would take the form:

s (s- 1) s (s- 1) (s - 2) f(n + s) = f(n) + s6f(n) +

2 6 2 f(n) +

6

6 3 f (n)

It is also interesting to note that since the time the sun's anomalous motion was incorporated into the calendar systems during the Sui and T'ang periods, the calendar-experts had not given their interpolation tedmiques any spe­cific names. It was loosely called the method of "mutual subtraction and

150

mutual multiplication" during the T'ang period 15 • However, with the use of the tedmique for the summation of power progressions in the Shou-shih calendar system, the Meth;d of Finite Differences went by the name Chao Ch'a 1451 (Calling the Difference) since the Yüan dynasty in the thirteenth century. Thereafter the chao ch'a method entered the main stream of tradi­tional mathematics and assumed its theoretical basis. In the hands of the mathematicians, the interpolation formula was systematically derived and applied to in a grea,t variety of problems. The famous Yüan mathematician Chu Shih-chieh 1461 extended the finite differences up to order four 16 and explained explicitly the procedure of the summation of power progressions. The formation of his interpolation formula, too, is identical with that formu­lated by Gregory-Newton in the seventeenth century.

15 The Ch'ung-wu calendar systems of the T'ang period, for example, menUans the phrase 'hsien }]siang chien hou hsiang ch'eng' 1441 (mutual subtraction before mutual multiplication) seven times in connection with the tedmique of interpolation adopted from the Ta-yen calendar system of 1-Hsing. See Hsin T'ang shu, eh. (B), pp. 2a, ff.

16 CHu Shih-chieh illustrated how the metbad of finite differences couJd be apptied in the last five problenis in chapter 2 of his Ssu yüan yü chien[41] (PreciOUli mirrar of the four elements) whic::h he wrote in 1303. One of the problems involves finite differen­ces up to the fourth order. Written in the modern form, the problern is solved by the

formula: f(n) = n~1 + - 1- n (n -I) ~2 + - 1- n(n -1) (n- 2} ~~ 21 31

+ ---J-r--n(n-1) (n-2) (n-3) ~·.

For an explanation of the solution, see Ho Peng-yoke, NChu Shih-chieh•, in Diction­ary o_f scientitic biography, vol. III, New York, 1971, pp. 269-270; CHIEN Pao-tsung, op. c1t. pp. 200-201; Li Yen, op. cit. pp. 7~4.

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