卷三·第 174–184 问　太阳均差、交点、本轮均轮与偏心等速圈

§174. What is the equation of the Sun?

第一百七十四问　何谓太阳的均差？

It is known that when the Sun is at either of the two points of apogee and perigee, the two lines drawn to it from the centre of the apogee orb and from the centre of the world coincide, so that there is no difference between them; and that when it is at any other place, the lines drawn to its body from the two said centres diverge. Their difference, in the way that comes nearest to a picture of it, is the equation. More precisely: it has been shown in the art of geometry that angles at the centre of a circle and at its circumference correspond to the arcs, since they are in proportion to them; so when the arcs traversed on the circumference of the apogee orb in equal times are equal, the angles standing on them at the centre are equal. For this reason it comes to the same thing, for the mean position, whether we say the arc of the Sun's distance in the apogee orb from the point in it facing the beginning of Aries, or the angle enclosed by the two lines drawn from the centre of the apogee orb, one of them to that point and the other to the Sun; and for the true portion, that it is the angle enclosed by the two lines drawn from the centre of the world, one of them to the apogee and the other to the Sun. The equation, then, is the difference between the two portions, the mean and the true, and its amount is the amount of the angle formed at the Sun by the two lines drawn to it, one of them from the centre of the apogee orb and the other from the centre of the world. This is the figure of it.

已知当太阳位于远地点、近地点两点中的任一点时，从远地点圈圆心与从世界中心引向它的两条线重合，两者之间没有差别；而当它在其余任何地方时，从上述两个圆心引向太阳本体的两条线就分开了。它们之间的差，用最接近图像的说法来讲，就是均差。说得更确切些：几何之术已经证明，圆心处与圆周处的角对应于弧，因为它们与弧成比例；所以当远地点圈圆周上在相等时间内走过的弧相等时，立在这些弧上的圆心角也相等。因此，就平位置而言，说它是太阳在远地点圈中距该圈上正对白羊起点之点的弧，或者说它是从远地点圈圆心引出的两条线——一条到那一点，另一条到太阳——所夹的角，都是一回事；就真份而言，它是从世界中心引出的两条线——一条到远地点，另一条到太阳——所夹的角。于是均差就是平均份与真份这两份之差，它的大小，就是从两处引向太阳的两条线——一条来自远地点圈圆心，另一条来自世界中心——在太阳处所成之角的大小。这是它的图。

§175. How great are the motions of the Sun in the sphere of the Sun?

第一百七十五问　太阳天球中太阳诸运动的大小如何？

The mean motion of the Sun in the order of the signs, in a day with its night, is 0;59,8,20, and its revolution through the whole zodiac takes approximately 365 days and 5 hours, a third, a quarter and a fifth of an hour [corrected: MS has 'a third and a quarter of a fifth of an hour']; this period is called the year of the Sun. Its apogee also moves in the order of the signs, 1 degree in every 66 years of the years of the Sun [corrected: MS has 'every year and sixty years'], according to the findings of the moderns; the ancients disagreed about it and about its amount. The distance between the centre of the world and the centre of the apogee orb is approximately 2 parts, when the radius of the apogee orb is 60 parts.

太阳顺星座次序的平均运动，一日一夜为零度五十九分八秒又二十个三次分；它走遍整个黄道一周约需三百六十五日又五小时，再加一小时的三分之一、四分之一与五分之一〔抄本作“三分之一又五分之一的四分之一小时”，据算改正〕；这段时间称为太阳年。它的远地点也顺星座次序移动，照近人的测定，每六十六个太阳年移动一度〔抄本作“每一年又六十年”，据算改正〕；古人对此以及它的数量意见不一。当远地点圈的半径为六十份时，世界中心与远地点圈圆心之间的距离约为二份。

§176. What is the inclined orb?

第一百七十六问　何谓斜圈？

The six planets do not keep to the zodiac in their course, but deviate from it, sometimes to the north and sometimes to the south, because their course is on orbs whose planes are inclined to the plane of the zodiac, just as the zodiac is inclined to the equator. The inclination of the inclined orb is not of one amount for all the planets, and the place of its greatest inclination from the zodiac also differs; but the centre of the parecliptic and of the inclined orb is one, namely the centre of the world.

六颗行星在运行中并不守着黄道，而是偏离它，时而偏北、时而偏南，因为它们所走的圈，平面倾斜于黄道的平面，正如黄道倾斜于赤道。斜圈的倾角并非所有行星都是同一个量，它离黄道倾斜最大之处也各不相同；但拟黄道圈与斜圈的圆心是同一个，就是世界中心。

§177. What is the jawzahr?

第一百七十七问　何谓“焦扎赫尔”？

When the plane of the inclined orb is inclined to the plane of the zodiac, the two orbs necessarily intersect at two opposite places, just as the zodiac intersects the equator at two opposite places. The name jawzahr applies to each of them; and when one wishes to distinguish one from the other, the intersection past which the planet passes towards the north is the Head, and the other intersection, past which it proceeds towards the south, is the Tail. When jawzahr, Head and Tail are used without qualification, they belong to the Moon; the others are attributed to the planets to which they belong. The Head is sometimes called the northern crossing and the northern node, and the Tail the southern crossing and the southern node. This is the figure of it.

斜圈的平面既然倾斜于黄道的平面，两圈必然在相对的两处相交，正如黄道与赤道在相对的两处相交。“焦扎赫尔”这个名称适用于其中每一处；若要把两者分开，行星经过它而向北去的那个交点是“龙头”，行星经过它而向南行进的另一个交点是“龙尾”。不加限定地说“焦扎赫尔”“龙头”“龙尾”时，指的是月亮的；其余的则归于它们所属的行星。“龙头”有时也称为北方的渡口、北方的结，“龙尾”称为南方的渡口、南方的结。这是它的图。

§178. What is the epicycle?

第一百七十八问　何谓本轮？

It is a small orb, not surrounding the earth, to which the planet keeps in the motion that is proper to it.

它是一个不环绕大地的小圈，行星在它固有的运动中守着这个圈。

§179. What is the deferent orb?

第一百七十九问　何谓均轮？

It is an orb whose centre is outside the centre of the world and whose plane is the plane of the inclined orb; it carries the epicycle on its circumference in the order of the signs. This is the figure of it.

它是一个圆心在世界中心之外、平面就是斜圈平面的圈；它把本轮载在自己的圆周上，顺星座次序运行。这是它的图。

§180. What is the equant orb?

第一百八十问　何谓偏心等速圈？

If the motion of the centre of the epicycle on the circumference of its deferent were by equal arcs in equal times, its mean course would be on it, and the angles at the centre of the deferent would also correspond to them. But these arcs of the deferent, and the angles at its centre, differ in equal times; and their equality is found at a point whose distance from the centre of the deferent, in each of Venus and the three upper planets, is equal to the distance of the centre of the deferent from the centre of the world, it and the two centres lying on one straight line. That point is the centre of the equant orb; and it is the same whether an orb equal to the deferent is drawn about it, on whose circumference the mean course of the centre of the epicycle is taken, or the mean motions are taken from the angles at its centre. This is the figure of it.

如果本轮中心在其均轮圆周上的运动，是在相等的时间内走过相等的弧，那么它的平均行程就在均轮上，均轮圆心处的角也会与之相应。但均轮上的这些弧，以及均轮圆心处的角，在相等的时间内并不相等；它们的相等是在某一点上见到的：在金星和三颗上行星中，这一点到均轮圆心的距离，等于均轮圆心到世界中心的距离，这一点与两个圆心在一条直线上。这一点就是偏心等速圈的圆心；绕着它画一个与均轮相等的圈，在它的圆周上取本轮中心的平均行程，或者从它圆心处的角取平均运动，都是一样的。这是它的图。

§181. What are the mean and the visible summit?

第一百八十一问　何谓平均顶点与可见顶点？

'Visible' in these matters is what is set against what is called 'mean'. The mean summit is the point at the top of the epicycle at which the line drawn from the centre of the equant through the centre of the epicycle ends. The visible summit is the point at the top of the epicycle at which the line drawn from the centre of the world through the centre of the epicycle ends. This is the figure of it.

在这些事情里，“可见”是与所谓“平均”相对而言的。平均顶点是本轮上端的一点，从偏心等速圈圆心穿过本轮中心所引的线终止于此。可见顶点是本轮上端的一点，从世界中心穿过本轮中心所引的线终止于此。这是它的图。

§182. What is the mean position of a planet?

第一百八十二问　何谓行星的平位置？

It is the distance of the centre of its epicycle from the point facing the head of Aries, according to the equant orb; that is, the angle enclosed by the two lines drawn from the centre of the equant to the head of Aries and to the centre of the epicycle.

它是行星本轮中心距正对白羊之首那一点的距离，按偏心等速圈来计；也就是从偏心等速圈圆心分别引向白羊之首与本轮中心的两条线所夹的角。

§183. What are the mean and the true anomaly?

第一百八十三问　何谓平近点角与真近点角？

The mean anomaly is the planet's distance in the epicycle from the mean summit, and the true anomaly is its distance from the visible summit; the difference between them is the equation of the first portion.

平近点角是行星在本轮中距平均顶点的距离，真近点角是它距可见顶点的距离；两者之差就是第一份的均差。

§184. What are the mean and the true longitude?

第一百八十四问　何谓平经度与真经度？

The mean longitude is the amount of the angle enclosed by the two lines drawn from the centre of the equant to its apogee and to the centre of the epicycle; the true longitude is the amount of the angle enclosed by the two lines drawn from the centre of the world to the apogee of the equant and to the centre of the epicycle. The difference between them is the equation of longitude, which is the amount of the angle formed by the said lines at the centre of the epicycle. Longitude is sometimes called the centre (markaz).

平经度是从偏心等速圈圆心分别引向其远地点与本轮中心的两条线所夹之角的大小；真经度是从世界中心分别引向偏心等速圈之远地点与本轮中心的两条线所夹之角的大小。两者之差就是经度均差，也就是上述两线在本轮中心处所成之角的大小。经度有时也称为“中心”。