卡里普斯Callippus of Cyzicus

卡里普斯的出生和死亡日期是猜测的，但已知他从公元前330年开始在雅典与亚里士多德一起工作。

我们知道卡里普斯是欧多克索斯学派的学生。我们还知道他在赫勒斯滂海峡沿岸进行了天文观测，这可以从观测本身推断出来。西里西亚的辛普利修斯在其对亚里士多德的De caelo的注释中写道（例如见[1]）：-

卡里普斯曾师从欧多克索斯的学生波勒马库斯，随后跟随他前往雅典，与亚里士多德同住，在亚里士多德的帮助下，修正并完善了欧多克索斯的发现。

卡里普斯精确测定了各季节的长度，并构建了一个76年周期，包含940个月，以协调太阳年和太阴年，该周期于公元前330年被采用，并为后来所有天文学家所使用。巴特尔·伦德特·范德瓦尔登在[6]中详细考察了卡里普斯的这一历法。克劳狄乌斯·托勒密在AlmagestⓉ（主要论题：源自阿拉伯语'al-majisti'——希腊语'Mathematike Syntaxis'的阿拉伯语翻译，后译为拉丁语'Magna Syntaxis'）中为我们提供了该周期于公元前330年开始的准确日期，称第一个周期的第50年与亚历山大去世后的第44年重合。

卡利普斯周期基于默冬（约公元前460年出生）设计的默冬周期。默冬的观测于公元前432年在雅典进行，但他给出的年长比实际多$\large\frac{1}{76}\normalsize$天。卡里普斯的周期与默冬周期之间的关系在[2]中解释如下：-

卡里普斯（约公元前370-300年）可能是他那个时代最杰出的天文学家。他构建了所谓的卡利普斯周期，本质上是由四个默冬周期组成的循环。它比原始的默冬周期更精确，并利用了365.25天是比365天更精确的回归年值这一事实。卡利普斯周期由4 × 235，即940个朔望月组成，但其大小月的分布与默冬不同。卡里普斯没有采用440个小月和500个大月的总数，而是采用了441个小月和499个大月，从而将四个默冬周期的长度减少了一天。因此所涉及的总天数变为(441 × 29) + (499 × 30)，即27,759，而27,759 ÷ (19 × 4)恰好给出365.25天。因此，卡利普斯周期将940个朔望月精确地拟合为76个回归年，每年365.25天。

卡里普斯引入了34个天球的系统来解释天体的运动。太阳、月亮、水星、金星和火星各有五个天球，而木星和土星各有四个，恒星有一个。与欧多克索斯提出的系统相比，增加了六个天球，提高了理论的准确性，同时保留了天体必须具有基于圆形的运动这一信念，因为那是“完美”的路径。托马斯·利特尔·希思写道[4]：-

卡利普斯试图通过增加其他球层，使同心球的体系更精确地符合现象；对于木星和土星，他保留球层数为四，但对其他行星各增加一个球层，对太阳和月亮各增加两个球层……这将用更复杂的拉长图形取代马鞍形曲线[见欧多克索斯条目]……

卡里普斯对数学天文学的其他贡献包括他观察到季节长度的不等性。他在模型中通过使太阳的速度在一年中变化来解释这一点，这是通过上述两个额外球层实现的。

卡利普斯周期有助于后来天文理论的精确性。Kieffer在[1]中写道：-

尽管同心球层体系让位于epicycles和eccentrics，卡里普斯的周期成为许多世纪中精确关联观测的标准，从而有助于后来天文理论的精确性。

The dates given for the birth and death of Callippus of Cyzicus are guesses but he is known to have been working with Aristotle in Athens starting in 330 BC.

We know that Callippus was a student in the School of Eudoxus. We also know that he made his astronomical observations on the shores of the Hellespont, which can be deduced from the observations themselves. Simplicius writes in his commentary on De caelo by Aristotle (see for example [1]):-

Callippus of Cyzicus, having studied with Polemarchus, Eudoxus's pupil, following him to Athens dwelt with Aristotle, correcting and completing, with Aristotle's help, the discoveries of Eudoxus.

Callippus made accurate determinations of the lengths of the seasons and constructed a 76 year cycle comprising 940 months to harmonise the solar and lunar years which was adopted in 330 BC and used by all later astronomers. This calendar of Callippus is examined in detail by van der Waerden in [6]. Ptolemy gave us an accurate date for the beginning of this cycle in 330 BC in the Almagest Ⓣ saying that year 50 of the first cycle coincided with the 44th year following the death of Alexander.

The Callippic period is based on the Metonic period devised by Meton (born about 460 BC). Meton's observations were made in Athens in 432 BC but he gave a length for the year which was $\large\frac{1}{76}\normalsize$ of a day too long. The relation between Callippus's period and that of Meton are explained in [2] as follows:-

Callippus of Cyzicus (c. 370-300 BC) was perhaps the foremost astronomer of his day. He formed what has been called the Callippic period, essentially a cycle of four Metonic periods. It was more accurate than the original Metonic cycle and made use of the fact that 365.25 days is a more precise value for the tropical year than 365 days. The Callippic period consisted of 4 × 235, or 940 lunar months, but its distribution of hollow and full months was different from Meton's. Instead of having totals of 440 hollow and 500 full months, Callippus adopted 441 hollow and 499 full, thus reducing the length of four Metonic cycles by one day. The total days involved therefore became (441 × 29) + (499 × 30), or 27,759 and 27,759 ÷ (19 × 4) gives 365.25 days exactly. Thus the Callippic cycle fitted 940 lunar months precisely to 76 tropical years of 365.25 days.

Callippus introduced a system of 34 spheres to explain the motions of the heavenly bodies. The Sun, Moon, Mercury, Venus and Mars each had five spheres while Jupiter and Saturn had four and the stars had one. This addition of six spheres over the system proposed by Eudoxus increased the accuracy of the theory while preserving the belief that the heavenly bodies had to possess motion based on the circle since that was the 'perfect' path. Heath writes [4]:-

Callipus tried to make the system of concentric spheres suit the phenomena more exactly by adding other spheres; he left the number of spheres at four in the case of Jupiter and Saturn, but added one each to the other planets and two each in the case of the sun and the moon ... . This would substitute for the hippopede [see the Eudoxus article] a still more complicated elongated figure ...

Other contributions of Callippus to mathematical astronomy included his observation of the inequality in the lengths of the seasons. He accounted for this in his model by making the velocity of the Sun vary through the year and this was achieved with the two extra spheres described above.

The Callippic period contributed to the accuracy of later astronomical theories. Kieffer writes in [1]:-

Although the system of concentric spheres gave way to epicycles and eccentrics, Callippus's period became the standard for correlating observations accurately over many centuries, and thus contributed to the accuracy of later astronomical theories.