1. Introduction

The Yajnavalkya 95-year cycle is a method of reconciling the lunar and solar calendars. It was proposed by the ancient Indian sage Yajnavalkya, who is believed to have lived around the 9th - 8th century BCE.^[1][2]

This  method of harmonization between the lunar and solar calendars is very important in making Indian Calendars.

The relative positions of the Sun and the Moon in the sky are constantly changing. This means that the positions of the Sun and the Moon relative to each other will gradually change over time. However, there is a pattern to this change. Every 95 years, the Sun and the Moon return to approximately the same positions in the sky at the same time of year. This is because 95 is the least common multiple of the synodic periods of the Sun and the Moon (the synodic period is the average time between two successive conjunctions of the Sun and the Moon).

The accuracy of the Yajnavalkya cycle is remarkable. After 95 years, the Sun and the Moon are only about 1° out of alignment. This is because the synodic periods of the Sun and the Moon are not exactly constant. They vary slightly due to the elliptical shapes of the orbits of the Earth and the Moon. However, the variations are small enough that the Yajnavalkya cycle remains accurate for centuries.

According to Eggeling, the same 95 Years of cycle was also mentioned by the school of Shandilya in the 10th-14th Kanda of Satapatha Brahmana, therefore it is also known as Shandilya Cycle.^[3]

2. Concept of the Cycle 

The lunar calendar is based on the cycles of the Moon, and has 12 months of 29.5 days each. This means that the lunar calendar is about 11 days shorter than the solar calendar, which is based on the Earth’s orbit around the Sun.

There is a logic behind this cycle that if the year is counted as 360 Tithis, then this leads to exactly 35 intercalary months (with a residual small error) in 95 years. Thithi means lunar day.

The Yajnavalkya 95-year cycle corrects for this difference by adding an extra month (Adhika Masa) to the lunar calendar every 32.5 years. This means that in a 95-year cycle, there will be 71 lunar years and 70 solar years.

Here is an example of how the Yajnavalkya 95-year cycle works:

If in the Yajnavalkya Year 1, the solar and lunar calendars are aligned.

Then in the year 2, the lunar calendar is 11 days behind the solar calendar.

In the year 32.5, an extra month is added to the lunar calendar to bring it into alignment with the solar calendar.

In the year 43.5, the lunar calendar is 11 days behind the solar calendar again.

In the year 65, an extra month is added to the lunar calendar to bring it into alignment with the solar calendar again.

In the year 95, the solar and lunar calendars are aligned again.

Yajnavalkya Year

Yajnavalkya took 360 days (Tithis) or 12 months (each month having 30 days) in a year to devise the concept of the synchronisation of the solar and lunar motions in the calander. This period of 360 days in a year is called as Yajnavalkya Year. Similarly the period of 95 Yajnavalkya Years make a Yajnavalkya Cycle. 

Adhika Masha

A lunar year is 354 days and a solar year is 365 days. In this way, there is a difference of 11 days between the two and in three years this difference becomes 33 days, which is called Adhika Masha or extra month. On the basis of astrology, to eliminate the difference of these 11 days of the lunar and solar year, arrangements have been made for more months in three years.^[4]

3. Metonic Cycle

Yajnavalkya's 95-year cycle consisted of five sub-cycles of 19 years each. The 19 year subcycle is presently called the Metonic cycle. The 19 year cycle was derived from the 95 year cycle.

4. Sources of the Theory 

The main sources of the theory and the model are Shatapatha Brahmana and Shukla Yajurveda. Yajnavalkya was Indian astronomer who studied about the motion of the Sun and mentioned these theories in his work Shatapatha Brahmana. He invented a method of reconciliating the lunar calendar and the solar calendar.^[5] He described the 95-year cycle to synchronize the motions of the sun and the moon.^[6] It is mentioned as 95 year “Agnichayana” in the 6th Kānda of Shatapatha Brahmana.^[7] The Yajnavalkya cycle is a testament to the astronomical knowledge of the ancient Indians. It is a complex and accurate system that has been used for centuries to predict astronomical events. It shows that the ancient Indians had a deep understanding of the motions of the heavenly bodies. According to Shatapatha Brahmana, while performing Yajna the process of intercalation is included as a prescription that 95 altars be built in a sequence defining the 95-years cycle.^[3]

5. Introduction to Yajnavalkya

Yajnavalkya was described as the greatest Brahmajnyani by all the sages at the function organised by the king Janaka in the ancient Mithila region at his court. The headquarter of  the Ancient Mithila University was the court of the King Janaka .^[8] Yajnavalkya was a great multidisciplinary scholar, philosopher, astronomer and theorist of the Vedic cosmology. He was running his own individual institution for the study of Vedas and cosmology at Jagban village in the Mithila region of the Madhubani district in the present Bihar state of India. This institution is known as Yajnavalkya Ashram . The Teachings of Yajnavalkya include the physical and metaphysical concepts of Space,  Time and Consciousness. Due to his multidisciplinary wisdom, Yajnavalkya is taken as an epitome of knowledge and is considered as the Father of the Indian Philosophy. The major Upanishads of the Indian philosophy are attributed to Yajnavalkya. The concept of Atman and Brahman was strengthened by Yajnavalkya in his works of Upanishads. Similarly the concept of space and time discussed with the woman philosopher Gargi Vachaknavi at the court of King Janaka is the earliest reference on the concept of space - time in the intellectual history of the world.