Chapter 3: Direction, Place, and Time

On a stony surface, made water-level, or upon hard plaster, made level, there draw an even circle, of a radius equal to any required number of the digits ( arnjula ) of the gnomon (qanku).

At its centre set up the gnomon, of twelve digits of the measure fixed upon ; and where the extremity of its shadow touches the circle in the former and after parts of the day,

There fixing two points upon the circle, and calling them the forenoon and afternoon points, draw midway between them, by means of a fish-figure ( timi ), a north and south line.

Midway between the north and south directions draw, by a fish-figure, an east and west line : and in like manner also, by

The east and west line is called the prime vertical ( sama - mandala) ; it is likewise denominated the east and west hour circle (unmandala) and the equinoctial circle ( vishuvanmandala ). The line drawn east and west through the base of the gnomon may be regarded as the line of common intersection at that point of three groat circles, as being a diameter to each of the three, and as thus entitled to represent them all. These circles are the ones which in the last figure (Fig. 8, j> . 101) are shown projected in their diameters ZZ' PP' and EE'; the centre C, in which the diameters intersect, is itself the projection of liio line in question here. ZZ' represents the prime vertical, which is styled sama mandala, literally “even circle: ” PP' is the hour circle, or circle, of declination, which passes through the east and west points of the observer’s horizon; it is called unmandala “ up-circle ” — that is to sav, the circle which in the oblique sphere is elevated; EE' finally, the equator, has the name of vishuvan mandala, or vishuvadvrtia, “ circle of the equinoxes;” the equinoctial points themselves being denominated vishuvat , or vishuva, which may be rendered ” point of equal separation.” 'The same line of the dial might be regarded as the representative in like manner of a fourth circle, that of the horizon ( kshiiija ), projected, in the figure, in SN : hence the commentary adds it also to the other three; it is omitted in the text, perhaps, because it is represented by the whole circle drawn about the base of the gnomon, and not by this diameter alone. The specifications of this verse, especially of the latter half of it, are of little practical importance in the treatise, for there hardly arises a case, in fxny of its calculations, in which the east and west axis of the dial comes to be taken as standing for these circles, or any one of them. In drawing the base ( bhuja ) of the shadow, indeed, it does represent the plane of the prime vertical (see below, under vv. 23-25); but this is not distinctly stated, and the name of the prime vertical (samamandala) occurs in only one other passage (below, v. 2G) : the east find west hour- circle ( unmandala ) is nowhere referred to again ; and the equator, as

*Draw likewise an east and west line through the extre- mity of the equinoctial shadow (vishuvudbhd) ; the interval be- tween any given shadow and the line of the equinoctial shadow is denominated the measure of amplitude ( ayra ). The equinoctial shadow is defined in a subsequent passage (vv. 12, 13); it is, as we have already had occasion to notice (under ii. 61-63). the shadow cast at mid-day when the sun is at either equinox — that is to say, when he is in the plane of the equator. Now as the equator is •> circle of diurnal revolution, the line of intersection of its plane with +hat of the horizon will be an east and west line; and since it is also a great circle, that line will pass through the centre, the place of the observer: if, therefore, we draw through the extremity of the equinoctial shadow a line parallel to the east and west axis of the dial, it will repre- sent the intersection with the dial of an equinoctial plane passing through the top of the gnomon, and in it will terminate the lines drawn through that point from any point in the plane; of the equator; and hence, it will also coincide with the path of the extremity of the shadow on the day of the equinox. Thus, let the following figure (Fig. 9) repre- sent the plane of the dial, NS and MW being its two axes, and b the base of the gnomon : and let the shadow cast at noon when the sun is upon the equator be. in a given latitude, be : then hr is the equinoc- tial shadow, and QQ', drawn through e and parallel to FAV, is the path of the equinoctial shadow, being the Jine in which a ray of the sun, from any point in the plane of the equator, passing through the top of the gnomon, will meet the face of the dial . Tn the figure as given, the circle is supposed to be described about the base of the gnomon with a radius of forty digits, and the graduation of the eastern and western sides of the circumscribing square, used in measuring the base ( bhuja ) of the shadow, is indicated:

The square root of the sum of the squares of the gnomon and fhadow is the hypothenuse : if from the square of the latter remainder is the shadow ; the gnomon is found by the converse process. :u

In an Age ( yitqa ), the circle of the asterisms (bha) falls back eastward thirty score of revolutions. Of the result obtained after multiplying the sum of days (dyuqana) by this number, and dicing by the number of natural days in an Age,

Take'the part which determines the sine, multiply it by three, and divide by ten ; thus are .found the degrees called those of the precession (aijana). From the longitude of a planet as corrected by these are to be calculated the declination, shadow, ascensional difference ( caradala ), etc.

The circle, as thus corrected, accords with its observed place at the solstice (ayana) and at either equinox ; it has moved eastward, when the longitude of the sun, as obtained by calcu- lation, is less than that derived from the shadow,

By the number of degree s of the difference ; then, turning back, it has moved westward by the amount of difference, when the calculated longitude is greater. . . . Nothing could well be more awkward and confused than this mode of stating the important fact of the precession of the equinoxes, of describing its mothod and rate, and of directing how its amount at any time is to be found The theory which the passage r in its present form, is actually intended to put forth is as follows * the vernal equinox librates westward and eastward from the fixed point, ie*u Piscium, assumed as the com- mencement of the sidereal sphere — the limits of the libratory movement being 27° m either direction from that point, and the time of a complete revolution of libration being the six-hundredth part of the period called the Great Age (see above, under i. 15-17), or 7200 years; so that the annual rate of motion of the equinox is 54", We will examine with some care the language in which this theory is conveyed, as important results are believed to be deducible from it. * The first half of verse 9 professes to teach the fundamental fact pf the motion in precession. The words bhdndm cakratn , which we have rendered “ circle of the asterisms/’ i.e , the fixed zodiac, would admit nf being translated “ circle of the signs/' i.e., the movable zodiac, reckoned from the actr tl equinox, since bha is used in this treatise in

Gives the sines of co-latitude ( lamba ) and of latitude ( aksha ) : the corresponding arcs are co-latitude and latitude, always south. . . . The proportions upon which these rules are founded are illustrated by the following figure (Fig 11), in which, as m a previous figure (Fig 8,^ p 101), ZS represents a quadrant oi tho meridian, Z being the zenith and 8 tl*c south point, 0 being the centre, and EC the projection of the plane of the equator In order to illustrate the corresponding relations of the dial, we have conceived the, gnomon, C 6, to be placed at the centre Then, when the sun is on the meridian and in tho equator, at E, th# shadow cast, which is the equinoctial shadow, is be, while Ce is the correspond- ing hypothenuse. But, by similarity of triangles,

And the product divided by the corresponding hypothe- mise, the result, converted to arc, is the sun’s zenith-distance (nata), in minutes : this, when the base is south, is north, and when the base is north, is south. Of the sun’s zenith-distance and Ins declination, in minutes,

is the sine of co-latitude. . . . This passage applies to cases in winch the sun is not upon the equator^, but has a certain decimation, of which the amount and direction are known. Then, from the shadow cast at noon, ma\ be derived Lis zenith- v distance when upon the meridian, and the latitude Thus, supposing the sun, having north declination ED (Fig 11), to be upon the meridian, at D: the shadow of the gnomon will be bd , and the proportion C d db CD : DB /,// gives DB"", the sine of the sun’s zenith-distance, ZD, which is found from it by the conversion of sine into are by a rule previously given (ii. 33). ZD in this case being south, and ED being north, their sum, EZ, is the latitude : if, the declination being south, the sun w r ere at D', the difference of ED' and ZD' would be EZ, the latitude. The ijgure dpcs not give an illustration of north zenith-distance, being drawn for the latitude of Washington, where that is impossible The latitude being thus ascertained, it is easy to find its sine and cosine : the only thing which deserves to bo noted in the process is that, to find the ^co- sine from the sine, resort is had to the laborious method of squares* instead of taking from the table the sine of the complementary arc, or -the lotijyd . , The sun’s distance from the zenith when he is upon the meridm njp .. called naids , “ deflocted,” an adjective belonging to the tiohn ItpfmC “ minutes,” or bhdgd s, anqds, " degrees.” The same terai employed, as will be seen farther on (vv. 34-36), to designate angle. For zenith-distance off the merld^n another term is. below, v. S3).

If their direction he different, is the sun’s declination : if the sine ol this letter he multiplied by radius and divided by the sine ol greatest decimal ion, the result, converted to arc, will be the sun’s longitude, il lie is in the quadrant commencing with Aries ;

If in that commencing with Cancer, subtract from a half- circle; il in that commencing vith Libra, add a half-circle; if in that commencing with Capricorn, subtract from a circle : the re- sult, in each case, is the true (sphula) longitude of the sun at mid-day.

To this il the equation ot the apsis (mdnda phala) be repeatedly applied, with a contrary sign, the sun’s mean longitude will be found. . . .

And divided by the perpendiculai-sinc, the results are ,the shadow and hypothenusc at mid-day The problem here is to determine the length ol the shadow which will be cast at mid day when the sun has i gnen dichnation the latitude of the observer being ilso known First tlic sun s meiidmn zenith distance is found by a pioccss the converse of th it t lu^lit in veises 15 and 16, then, the coifespondmg sine and cosine lining been eilculated i Simple propoition gives the desired loult Thus let us suppose tlic sun to be at I)' (Fig 11 p 121), the sum of lus south declination ED', and the north latitude, EZ, gives the zenith ehst mce ZD' its sine (bhujajyd) is D'B'", and its co«ine ( hotijyd ) is ( B'" llicn CB'" B'"D' Lb bd 1 and CB'" CD' Cb CcV which proportions, reduced to equations grv^ (he v due of bd 1 , the shadow, and Cd', its hepothenuse 22 . The sine of decimation, multiplied by tile equinoc- tial hypothenusc, and divided by tlic gnomon-sine ((^anlxupia), 23 Gives, when faithei multiplied by the hjpothenuse of any given shadow, and divided by ladms (madhyakarna), the sun’s measure of amplitude ( arldgrd ) conesponding to that shadow . . . In this passage we are taught the declination being known how to find the measure of amplitude (ugui) of am given shadow, is preparatory to de tcimming, by the next following rule the b ise (bhujo) of the shadow, by calculation instead of measurement r Ihe fust step is to find the suite of the suns amplitude in order to this, we compaio the triangles ABC and CEH (Fig 13, p 126), w/hich uc simiJ u, since the angles ACB and CE0 are e ich equal to the latitude of the obseiver Hence EH EC BC AC But the triangles CEH (Fig 13) and Cbc (Fig 11) are also smi&r, and EH EC Cb Ce Hence, b) equality of latios, Cb Ce BC AC ^ and AC, the sine of the sun’s amplitude, equals BC — which is the of declination, being equal to DF — multiplied bv Ce, the equinoctial* hjpothenuse, and divided by Cb the gnomon The remaining part ^of ; the process depends upon the principle which w r e have demon strated'^^eg. under verse 7, that the measure of amplitude is to the hvpothenuse shadow as the gjne of amplitude to radius

. . . The sum ol tlie equinoctial shado^ and the sun*s me asm e of amplitude (arhdgm), when the sun is m the southern hemisphere, is the base, north ; * 24 When the sun is in the northern hemisphere, the base is found, if noi th, by subtracting the measure of amplitude from the equinoctial shadow ; iJ south, by a contrary process — accord- ing to the direction of the intei val between the end of the shadow and the east and west axis 25 The mid-day base is mvaiiably the midday shadow . . . We hive already hac 1 occasion to notice m connection with the first verses of this clnpicr th it the base (bhuja) of the shadow is its projec- tion upon a noitli and south line, or the distance of its extremity from the cast and west axis of the dial It is that line which, as shown above (under v 7) conesponds to uid repiesents the perpendicular let fall from the sun upon the plane of the prime vertical Thus, it (Fig 11, p 121) K, L, D', D be different positions of the sun — K md L being conceived to be upon the surface of the spheie — the perpcnchculais KB', LB", D'B"', DB"" are iopr< suited upon the di il by 7T>, lb, d'b db, or, m Fig 9 (p 111) by 7 vb f lb " r d'b db Of these, the two Irttei coincide with their respective shadows, the shadow cast at noon being always itself upon a north and south line r lhe base of an^ shadow mi\ be found by pombming its measure of amplitude (a<pa) with the equinoctial shadow When the sun is m the southern hemisphere, as at I)' oi Iv lg 11) the measure of amplitude, td' or ch is to be added always to the equinoctial shadow, be , in ordei to give the base, bd oi bh If, on the contrary, the suit’s decimation bo noith x different method of proceduie will be necessary, according a&he is north oi south fiorri the prime \ertical If he be*bouth, as at D, the shadow, bd will be thrown northward, md the base will be found by subtracting the measure of amplitude, dc, from the equinoctial shadow, bt if he be north, as at L the cxtiumt} of the shadow, l , will Jate $Outh from the east and west axis, and the base, 67, will be obtained t subtracting the equinoctial shadow, be, from the measure of ampli- le , 25'. . . . Multiply the sines of co-latitude and of latitude re- stively by the equinoctial shadow and by twelve, •£;*' “26. And divide by the sine of declination ; the results are the ^gothenuse when the sun x| r on the prime vertical (samctmandala)'.

Multiplied by the equinoctial shadow, and divided by the mid-day measure of amplitude (agrd), is the hypothenuse. . . . He^e we have two separate and independent methods of finding the * hypothenuse of the oust and west shadow cast by the sun at the moment when he is upon the prime vertical. In connection with the second of the two are stated the circumstances under which alone a transit of the sun across the prime vertical will take place* if his decimation is south, or if, being north, ^it is greater than the latitude, his diurnal revolution will be wholly to the south, or wholly to the north, of th.it circle.

And this, being faithei multiplied by the hypothenuse ot a given shadow at that time, and divided by radius, gives the measuic ol amplitude (a<jtd), in digits (anqula), etc. . . . The sine of the sun’s amplitude is found — Ins decimation and the latitude being known — by a comparison ol the similar tuongles ABC and CEH (Fig 13), m which JIE EC BC CA or ^ cos lat It sm decl sm ampl And the proportion upon which is founded the lulc m veise 28 — namely, that radius lo the sine of amplitude as the hypothenuse of a given shadow to the corresponding measure of amplitude — has been demonstrated under versto 7, above * 28* ... If from hall the square ol radius the square of the %tne iCjf amplitude ( a q raj yd ) be subtracted, and the remainder multi*

And again multiplied by twelve, and then farther divided t>y the square of the equinoctial shadow increased by half the,, offthe gnomon— the result obtained by the wise

And divide as befoie the result is styled the “ fruit ” (phala). Add its squaie to the “ smd,” and take the squaie root of theif sum , tins, diminished and mci cased b} the <c fruit,” for the southern and northern hemispheres, 32 Is the sine of altitude (f anlu ) ol the southern inter- mediate dnections ( inhc ) , and equall}, whether the sun’s solu- tion take place to the south or to the noith of the gnomon (^anlii) * — only, m the latter case, the sine of altitude is that of the north- ern intermediate dnections 33 The square loot of the difference of the sqmies of that and of ladius is styled the sine ol zenith-distance (c/r() If, then, the sine of zenith-distance and ladius be multiplied lesjxetively by twehe, and dmded by the sine of altitude, 34 The lesults aie the shadow and hypothenuse at the angles ( Kona ), undei the given circumstances of time and place . . . The method taught m this passage of finding with a given decima- tion + and lihtude, the sine of the sun’s altitude it the moment when he crosses the south east and south west \crticdl circles or when the shadow of the gnomon is thrown toward the mgles (kona) of the circumscribing square of the dial, is, when st ited dgebraically, as follows

The result is the day-measure ( antyd ) ; this, diminished by the versed sine (uihrainajijd) of the hour-angle ( nata ), then multiplied by the day-radius and divided by radius, is the “ divisor ” ( chcda ) ; the latter, again, being multiplied by the sine of co-latitude (Jamba), and divided

By radius, gives the sine of altitude (<;anku) : subtract its sine from that of radius, and the square root of the remainder is the sine of zenith-distance ( (Jrr) : the shadow and its hypothenuse are found as in the preceding process. The object of this process is, to find the sine of the sun's altitude at any given hour of the day, when his distance from the meridian, his declination, and the latitude, are known. The sun’s angular distance

Is the sine of altitude (( atiku ) ; which, multiplied by radius and divided by the sine of co-latitude ( lam bn ), gives the “ divisor ” (cheda) ; multiply the latter by radius, and divide by the radius of the diurnal circle,

And the quotient is the sine of the sun’s distance from the horizon (unnala) ; this, then, being subtracted from the day- measure ( antyd ), and the remainder turned into arc by means of the table of versed sines, the final result is the hour-angle ( nata ), in respirations (asu), cast or west. The process taught in these verses is precisely the converse of the one described in the preceding passage. The only point which calls for further remark in connection with it is, that the line GQ (Fig. 16) is in verse 39 called the “ sine of the mma t a .” B\ this latter term is desig- nated the opposite of the hour-angle (nata) — that is to say, the sun’s angular distance from the horizon upon his own circle, O'A', reduced to time, or to the measure of a great circle. Thus, when the sun is at O', his hour-angle (nata), or the time till noon, is Q'E; his distance from the horizon (unnata), or the time since sunrise, is Q'G'. But GQ is with no propriety styled the sine of O'Q'; it is not itself a sine at all, and the actual sine of the arc in question would have a very different value.

Multiply the sine of co-latitude by any given measure of amplitude (agrd), and divide by the corrcsjjoiHliiig hypothenuse in digits; the result is the sine of declination. This, again, is to he multiplied by radius, and divided by the sine of greatest declination ;

The quotient, converted into are, is, in signs, etc., the sun’s place in the quadrant ; by means of the quadrants is then found the actual longitude of the sun at that point. . . .

From the point of intersection of the lines drawn between them by means of two fish-figures (matsya), and with a radius touching the three points, is described the path of the shadow. . . . This method of drawing upon the lace of the dial the path which will he described by the extremity of the shadow upon a given day proceeds upon the assumption that that path will be an arc of a circle — an erroneous assumption, since, excepting within the polar circles, the path of the shadow is always a hyperbola, when the sun is not in the equator. In low latitudes, however, the difference between the are of the hyperbola, at. any jToint not too far from the gnomon, and the arc of a circle, is so small, that it is not very surprising that the Hindus should have overlooked it. The path being regarded as a true circle, of course it can be drawn if' any three points in it can be found by calculation : and this is not difficult, since the rules above given furnish means of ascertaining, if the sun’s declination and the observer’s latitude be known, the length of the shadow. and Hie length of its base, or # the distance of its extremity from the east and west axis of the dial, at different times during the day. One part of the process, how- ever, has not been provided for in the rules hitherto given. Thus (Fig. 9, p. Ill), supposing (/, m , and / to be three points in tb* same daily path of the shadow, we require, in order to lay down / and tn, to know not only the bases lb", mb'", but also the distances hb ", bb"' . But these are readily found when the shadow and the base corresponding to each are known, or they may be calculated from Ihe sines of the respective hour-angles. The three points being determined, the mode of describing a circle through them is virtually the same with that which wo should employ : lines are drawn from the noon-point to each of the others, which are then, by fish-figures (see above, under vv. 1-5), bisected by other lines at right

In succession, the sines of one, of two, and of three sigi^s ; the quotients, converted into are, being subtracted, each from the one following, give, beginning with Aries, the times of rising (udayasaras) at Lanka ;

Namely sixteen hundred and seventy, seventeen hundred and ninety-five, and nineteen hundred and thirty-five respirations. And these, diminished each by its portion of ascensional differ- ence (carahhanda), as calculated for a, given place, are the times of rising at that place.

Invert them, and add their own portions of ascensional difference inverted, and the sums are the three signs beginning with Cancer : and these same six, in inverse order, are the other six, commencing with Libra. The problem here is to determine Ihe “ times of rising ” (udaydsavas) of the different signs of the ecliptic — that is to say, the part of the 5400 respirations (asavas) constituting a quarter of the sidereal day, which each of the three signs making up a quadrant of the ecliptic will occupy in rising ( udaya ) above the horizon. And in the first place, the times of rising at the equator, or in the right sphere — which are the equivalents of the signs in right ascension — are found as follows : Let ZN (Fig. 17) be a quadrant of the solstitial colure, AN the pro- jection upon its plane of the equinoctial colure, AZ of the equator, and AC of the ecliptic; and let A, T, G, and C be the projections upon AC of the initial points of the first four signs; then AT is the sine of one sign, or of 30°, AG of two signs, or of 60°, and AC, which is radius, the sine of three signs, or of 90°. From T, G, and C, draw Tf, Gg, C c, perpendicular to AN. Then ATf and ACr are similar triangles, and, since AC equals radius, It : Cc : : AT : T t But the arc of which T / is sine, is the same part of the circle of diurnal revolution of which the radius is //', as the required ascensional equivalent of one sign is of the equator; hence the sine of the latter, which we may call x , is

From the longitude of the sun at a given time are to be calculated the ascensional equivalents of the parts past and to come of the sign in which he is : they are equal to the number of degrees traversed and to be traversed, multiplied by the ascen- sional equivalent (udaydsavas) of the sign, and divided by thirty ;

Then, from the given time, reduced to respirations, sub- tract the equivalent, in respirations, of the part of the sign to come, and also the ascensional equivalents ( lagndsavas ) of the following signs, in succession — so likewise, subtract the equiva- lents of the part past, and of the signs past, in inverse order ;

So, from the east or west hour-angle ( nata ) of the sun, in nadls, having made a similar calculation, by means of the equiv- alents in right ascension (lankodaydsavas) , apply the result as an additive or subtractive equation to the sun’s longitude : the result is the point of the ecliptic then upon the meridian (madhyalagna ) . The word lagna means literally “ attached to, connected w T ith,” and hence, “corresponding, equivalent to.” It is, then, most properly, and likewise most usually, employed to designate the point or the arc of the equator which corresponds to a given point or arc of the ecliptic. In such a sense it occurs in this passage, in verse 47, where lagndsavas is precisely equivalent to udaydsavas , explained in connection with the next preceding passage; also below, in verse 50, and in several other places. In verses 48 and 49, however, it receives a different signification, being taken to indicate the point of tlie ecliptic which, at a given time, is upon the meridian or at the horizon; the former being called lagnam kshitije , “ lagna at the horizon or> i n one or two cases elsewhere, simply lagna — the other receiving the name of madhyalagna , “ meridian-lagna. The rules by which, the sun’s longitude and the hour of the day being known, the points of the ecliptic at the horizon and upon the meridian are found, are very ellipticaily and obscurely stated in the text; our translation itself has been necessarily made in part also a paraphrase and explication of them. Their farther illustration may be best effected by means of an example, with reference to the last figure (Fig. 18). At a given place of observation, as Washington, let the moment of local time— reckoned in the usual Hindu manner, from sunrise^-be 18"

Add together the ascensional equivalents, in respirations, of the part of the sign to be traversed by the point having less longitude, of the part traversed by that having greater longitude, and of the intervening signs — thus is made the ascertainment of time ( kalasddhana ).

When the longitude of the point of the ecliptic upon the horizon (lagna) is less than that of the sun, the time is in the latter part of the night ; when greater, it is in the day-time ; when greater than the longitude of the sun increased by half a revolution, it is after sunset. The process taught in these verses is, in a manner, the converse of that which is explained in the preceding passage, its object being to find the instant of local time when a given point of the ecliptic will be upon the horizon, the longitude of the sun being also known. Thus (Fig. 18), supposing the sun’s longitude, AP, to be, at a given time, I s 12°; it is required to know at what time the point H, of which the longitude is 4* 25°, will rise. The problem, is, virtually, to ascertain the arc of the equator intercepted between p, the point which rose with the sun, and h, wdiich will rise with H, since that arc determines the time elapsed between sunrise and the rise of H, or the time in the day at which the latter will take place. In order to this, we ascertain, by a process similar to that illustrated in connection with the last passage, the bhogyasavas , “ ascensional equivalent of the part of the sign to be traversed,” of the point having less longitude — or pg — and the bhukt&8av(i8 t '* ascensional